Integral Steel Box-Beam Pier Caps (2004)

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

I-1 APPENDIX I DESIGN EXAMPLE TABLE OF CONTENTS I1 PROBLEM STATEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-6 I2 NOTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-6 I3 GENERAL OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-9 I4 DESIGN PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-12 I5 PRELIMINARY DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-12 I5.1 I-Girders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-12 I5.2 Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-13 I5.3 Pier Cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-13 I6 COMPUTER MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-14 I6.1 Description of Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-15 I6.1.1 Girders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-15 I6.1.2 Slab and Cross Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-17 I6.1.3 Pier Cap and Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-18 I7 DEAD LOAD ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-18 I8 SEISMIC ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-19 I8.1 Seismic Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-19 I8.2 Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-19 I8.3 Equivalent Transverse Static Earthquake Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-20 I8.4 Equivalent Longitudinal Static Earthquake Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-21 I8.5 Intermediate Pier Column Seismic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-21 I8.6 Evaluate Slenderness Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-21 I8.7 Moment Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-22 I9 COLUMN DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-24 I9.1 Column Dead Load Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-24 I9.2 Column Live Load Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-24 I9.3 Load Cases for Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-24 I9.3.1 Load Cases for Extreme Event I Load Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-25 I9.3.2 Load Cases for Strength I Load Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-27 I9.4 Longitudinal Reinforcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-27 I9.4.1 Controlling Load Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-27 I9.4.2 Development Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-28 I9.5 Column Overstrength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-29 I9.6 Spiral Reinforcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-31

I9.6.1 Design Shear Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-31 I9.6.2 Shear Resistance of the Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-32 I9.6.3 Spacing of Spiral Reinforcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-33 I9.7 Column Reinforcing Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-34 I10 BEAM DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-36 I10.1 Earthquake Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-36 I10.2 Live Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-37 I11 CAP BEAM DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-37 I11.1 Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-37 I11.1.1 Factored Design Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-37 I11.1.2 Nominal Flexural Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-38 I11.1.3 Web Slenderness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-39 I11.2 Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-39 I11.2.1 Shear Forces for the Extreme Event I Load Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-39 I11.2.2 Shear Forces for the Strength I Load Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-40 I11.2.3 Nominal Shear Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-41 I11.3 Check of Box-Beam Flanges for Combined Moment and Torsional Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . I-42 I11.3.1 Extreme Event I Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-42 I11.3.2 Strength I Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-43 I11.4 Fatigue Requirements for Webs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-43 I11.5 Constructability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-44 I11.6 Service Limit State Control of Permanent Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-44 I11.7 Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-44 I12 GIRDER-TO-CAP BEAM CONNECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-45 I12.1 Bolted Double-Angle Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-45 I12.1.1 Shear Forces Due to Unfactored Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-45 I12.1.2 Slip Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-46 I12.1.3 Shear Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-46 I12.1.3.1 Design Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-46 I12.1.3.2 Nominal Bolt Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-48 I12.1.4 Bearing Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-48 I12.1.5 Size Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-49 I12.2 Flange Splice Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-49 I12.2.1 Girder Moments Due to Unfactored Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-49 I12.2.2 Size Flange Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-50 I12.2.2.1 Design Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-50 I12.2.2.2 Plate Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-50 I12.2.3 Design Connection to Girder Flanges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-51 I12.2.3.1 Slip Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-51 I12.2.3.2 Shear Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-52 I12.2.3.3 Bearing Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-52 I-2

I12.2.4 Design Connection to Cap Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-52 I12.2.4.1 Design Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-52 I12.2.4.2 Shear Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-54 I12.2.4.3 Slip Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-55 I12.2.4.4 Bearing Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-55 I12.3 Girder-to-Cap Beam Connection Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-55 I13 COLUMN-TO-CAP BEAM CONNECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-55 I13.1 Shear Studs on Bottom Flange Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-56 113.1.1 Strength Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-56 I13.1.1.1 Design Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-56 I13.1.1.2 Nominal Shear Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-57 I13.1.2 Fatigue Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-58 I13.1.3 Shear Stud Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-58 I13.2 Shear Studs on Web Plates of Cap Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-59 I13.2.1 Strength Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-59 I13.2.1.1 Shear Forces for Extreme Event I Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-59 I13.2.1.2 Shear Forces for Strength I Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-60 I13.2.1.3 Nominal Shear Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-60 I13.2.2 Fatigue Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-60 I13.2.2.1 Live Load Shear Force Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-60 I13.2.2.2 Fatigue Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-61 I13.2.3 Shear Stud Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-61 I13.3 Shear Studs on Diaphragm Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-61 I13.3.1 Strength Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-62 I13.3.1.1 Shear Force for Extreme Event I Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-62 I13.3.1.2 Shear Force for Strength I Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-62 I13.3.1.3 Nominal Shear Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-62 I13.3.2 Fatigue Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-62 I13.3.3 Shear Stud Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-63 I-3

LIST OF FIGURES Figure I-1. Bridge Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-10 Figure I-2. Bridge Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-10 Figure I-3. Girder Moments and Pier Cap Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-11 Figure I-4. Pier Cap Torsion and Column Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-11 Figure I-5. Girder-to-Cap Beam Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-11 Figure I-6. Splice Plate and Splice Plate Connection Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-12 Figure I-7. Shear Studs for Column-to-Cap Beam Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-13 Figure I-8. Shear Forces Acting on Shear Studs Located on Web Plates of Cap Beam . . . . . . . . . . . . . . . . . . . . . . . . . I-14 Figure I-9. Shear Forces Acting on Shear Studs Located on Internal Diaphragms of Pier Cap . . . . . . . . . . . . . . . . . . . I-14 Figure I-10. Preliminary Girder Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-15 Figure I-11. Preliminary Cap Beam Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-16 Figure I-12. SAP 2000 Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-16 Figure I-13. Modeling Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-17 Figure I-14. Column Dead Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-25 Figure I-15. Interaction Diagram for Factored Resistance at Bottom of Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-28 Figure I-16. Interaction Diagram for Factored Resistance at Top of Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-28 Figure I-17. Development Length for Column Longitudinal Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-29 Figure I-18. Interaction Diagram for Nominal Resistance at Top of Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-30 Figure I-19. Interaction Diagram for Nominal Resistance at Bottom of Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-30 Figure I-20. Free Body Diagram of Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-31 Figure I-21. Column Reinforcement Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-35 Figure I-22. Cap Beam Shear Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-39 Figure I-23. Shear Forces on Cap Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-41 Figure I-24. Detail Categories for Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-45 Figure I-25. Unfactored Torsion Diagrams for Pier Cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-53 Figure I-26. Girder-to-Cap Beam Connection Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-56 Figure I-27. Column-to-Cap Beam Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-57 Figure I-28. Stud Layout for Bottom Flange Plate of Cap Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-58 Figure I-29. Stud Layout for Web Plates of Cap Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-61 Figure I-30. Shear Stud Layout for Diaphragm Plates Adjacent to Joint Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-63 I-4

LIST OF TABLES Table I-1. Element Section Properties for Computer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-17 Table I-2. Noncomposite Dead Load Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-18 Table I-3. Noncomposite Dead Load Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-19 Table I-4. Uniform Dead Loads, N/mm (kip/ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-20 Table I-5. Column Seismic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-22 Table I-6. Magnified Column Moments for Earthquake Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-24 Table I-7. Unfactored Column Live Load Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-26 Table I-8. Load Cases for Extreme Event I Load Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-26 Table I-9. Load Cases for Strength I Load Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-27 Table I-10. Maximum Elastic Seismic Moments within the Girders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-36 Table I-11. Maximum Factored Girder Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-36 Table I-12. Maximum Unfactored Live Load Girder Moments, kN-m (k-ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-37 Table I-13. Maximum Elastic Seismic Pier Cap Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-40 Table I-14. Elastic Seismic Girder Shears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-46 Table I-15. Unfactored Girder Moments at Centerline of Pier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-49 I-5

I1 PROBLEM STATEMENT A design example for a bridge consisting of a steel I-girder superstructure integral with a steel box-beam pier cap supported on a single reinforced concrete column is presented. The steel box-beam pier cap and the concrete column are integrally connected by extending the longitudinal bars of the column into the pier cap compartment directly above the column and filling this compartment with concrete. The design is in accordance with the 1998 AASHTO LRFD Bridge Design Specifications, Second Edition, with 1999 through 2002 Interim Revisions, hereafter collectively referred to as the AASHTO LRFD Specifications. The design example is presented in SI units with equivalent U.S. customary units in parentheses. References to articles, equations, tables, and figures within the AASHTO LRFD Spec- ifications are made throughout the example and have been placed in bold print. Of particular interest and the main focus of this example is the design of the connection of the cap beam to the girders and column. The level of detail provided is that required by practicing engineers for design of such structures. I2 NOTATIONS A = seismic acceleration coefficient Ab = area of an individual bar (mm2); cross-sectional area of a bolt (mm2) Ag = gross cross-sectional area of a member (mm2) An = net cross-sectional area of a member (mm2) Ao = enclosed area within a box section (mm2) Asc = cross-sectional area of a stud shear connector (mm2) Asp = cross-sectional area of spiral reinforcing (mm2) Av = area of shear reinforcement (mm2) b = width of deck represented by a slab element (mm); compression flange width between webs (mm); width of member (mm) bv = effective web width, or for circular sections, the diameter of the section (mm) C = ratio of the shear buckling stress to the shear yield strength Csm = dimensionless elastic seismic response coefficient c = distance from the neutral axis to the outer fiber (mm) D = column diameter (mm); web depth (mm); width or depth of plate between webs or flanges (m) DC = designation for dead load due to structural components and nonstructural attachments Dc = depth of web in compression in the elastic range (mm) Dr = diameter of the circle passing through the centers of the longitudinal reinforcement (mm) DW = designation for dead load due to wearing surfaces and utilities d = nominal diameter of a bolt (mm); depth of pier cap (m); diameter of a shear stud (mm) db = nominal diameter of a reinforcing bar (mm) dc = outside diameter of spiral (mm) de = effective depth from extreme compression fiber to the centroid of the tensile force in the tensile rein- forcement (mm) do = stiffener spacing (mm) dv = effective shear depth (mm) E = modulus of elasticity (MPa) Ec = modulus of elasticity of concrete (MPa) EI = flexural stiffness (N-mm2) EQ = designation for earthquake load Fn = nominal flexural resistance in terms of stress (MPa) Fr = factored flexural resistance in terms of stress of the flange for which fu was determined (MPa) Fu = specified minimum tensile strength of steel (MPa); specified minimum tensile strength of a stud shear connector (MPa) Fub = specified minimum tensile strength of a bolt (MPa) Fy = specified minimum yield strength of steel (MPa) Fyc = specified minimum yield strength of the compression flange (MPa) Fyw = specified minimum yield strength of the web (MPa) fc = stress in the compression flange due to the factored loading under investigation (MPa) fc´ = specified compressive strength of concrete at 28 days (MPa) I-6

fcf = maximum compressive elastic flexural stress in the compression flange due to the unfactored perma- nent load and the fatigue load (MPa) fcw = maximum compressive flexural stress in the web (MPa) fu = flexural stress in the compression or tension flange due to the factored loading, whichever flange has the maximum ratio of fu to Fr in the panel under consideration (MPa) fy = specified minimum yield strength of reinforcing bars (MPa) g = acceleration of gravity (m/s2) H = horizontal shear (kN) HDSGN = design horizontal shear force (kN) HEQ = horizontal shear force due to seismic load (kN) HEXTR. EVENT I = horizontal shear force from Extreme Event I load combination (kN) HLL = horizontal shear force due to live load (kN) HSTR. I = horizontal shear force from Strength I load combination (kN) h = column height (m); height of a shear stud (mm) I = moment of inertia (mm4) Ig = moment of inertia of the gross concrete section about the centroidal axis (mm4) Inc = moment of inertia of the non-composite steel section (mm4) J = torsional inertia (mm4) K = bridge lateral stiffness (N/mm); effective length factor for compression members Kh = hole size factor for bolted connections K
u/r = slenderness ratio Ks = surface condition factor for bolted connections k = plate buckling coefficient; shear buckling coefficient; elastic bend-buckling coefficient for the web L = total length of bridge (mm) Lc = clear distance between holes or between the hole and the end of the member in the direction of the applied bearing force (mm) LL = designation for vehicular live load
d = development length (mm)
db = basic development length for straight reinforcement to which modification factors are applied to determine
d (mm)
u = unsupported length of a compression member (mm) M = moment (kN-m) MBOTT. = moment at bottom of column (kN-m) Mc = factored moment, corrected to account for second-order effects (kN-m) MCL = girder moment at centerline of pier cap (kN-m) MDC = unfactored moment due to structural components and nonstructural attachments (kN-m) MDC1 = unfactored moment due to DC loads applied to the non-composite steel section (kN-m) MDC2 = unfactored moment due to DC loads applied to the long-term composite section (kN-m) MDSGN = design moment for flange splice plates (kN-m) MDW = unfactored moment due to wearing surfaces and utilities (kN-m) MELASTIC = elastic seismic moment (kN-m) +MELASTIC = elastic seismic moment for the positive moment section of a girder (kN-m)
MELASTIC = elastic seismic moment for the negative moment section of a girder (kN-m) MEQ = moment due to seismic load (kN-m) ML = moment in the longitudinal direction or about the transverse axis of the bridge (kN-m) MLEQ = moment due to longitudinal earthquake load (kN-m) MLL = moment due to live load (kN-m) MMOD. = modified design moment (kN-m) Mn = nominal moment resistance (kN-m) MOVRSTR. = column overstrength moment resistance associated with plastic hinging of the column (kN-m) MSERV. II = moment from Service II load combination (kN-m) MT = moment in the transverse direction or about the longitudinal axis of the bridge (kN-m) MTEQ = moment due to transverse earthquake load (kN-m) MTOP = moment at top of column (kN-m) Mu = factored design moment (kN-m) I-7

M2b = moment on compression member due to factored gravity loads that result in no appreciable sidesway calculated by conventional first-order elastic frame analysis; always positive (kN-m) M2s = moment on compression member due to factored lateral or gravity loads that result in sidesway, ∆, greater than
u/1500, calculated by conventional first-order elastic frame analysis; always positive (kN-m) Ns = number of slip planes per bolt; number of shear planes per bolt P = axial load (kN) PDC = unfactored axial dead load due to structural components and nonstructural attachments (kN) PDL = axial dead load (kN) PDSGN = design force for flange splice plates (kN) PDW = unfactored axial dead load due to wearing surfaces and utilities (kN) Pe = Euler buckling load (kN) PLL = axial live load (kN) Pn = nominal compressive axial resistance and columns and nominal tensile resistance and splice plates (kN) PSERV. II = force from Service II load combination (kN) Pt = minimum required bolt tension (N) Pu = factored axial load (kN) pe = equivalent uniform static seismic loading per unit length of bridge applied to represent the primary mode of vibration (N/mm) po = a uniform load arbitrarily set equal to 1.0 (N/mm) Qn = nominal shear strength of a shear connector (kN) q = shear flow (kN/m) R = seismic response modification factor; shear interaction factor Rb, Rh = flange stress reduction factors Rn = nominal resistance of bolt, connection, or connected material (kN) Ru = factored force on bolt, connection, or connected material (kN) r = radius of gyration (mm) S = coefficient related to site conditions for use in determining seismic loads; elastic section modulus (mm3); spacing between interior beams (m) Snc = Section modulus of the non-composite steel section (mm3) s = spacing of spiral reinforcing (mm); bolt spacing (mm) T = torsion (kN-m) TEQ = torsion due to seismic load (kN-m) TLL = torsion due to live load (kN-m) Tm = period of bridge (s) Tu = factored torsion (kN-m) t = deck thickness (mm); plate thickness (mm); thickness of the thinner outside plate or shape (mm) tf = compression flange thickness (mm) tSPLICE PL. = splice plate thickness (mm) tw = web thickness (mm) U = reduction factor for shear lag V = shear force (kN) VAXIAL = shear force resulting from column axial load (kN) Vc = nominal shear resistance of the concrete (kN) VDC = unfactored shear due to structural components and nonstructural attachments (kN) VDC1 = unfactored shear due to DC loads applied to the non-composite steel section (kN) VDC2 = unfactored shear due to DC loads applied to the long-term composite section (kN) VDL = shear due to dead load (kN) VDSGN = design shear force for the connection (kN) VDW = unfactored shear due to wearing surfaces and utilities (kN) VEQ = shear due to seismic load (kN) VEXTR. EVENT I = shear from Extreme Event I load combination (kN) Vf = shear due to flexure (kN) VLEQ = shear due to longitudinal earthquake load (kN) VLL = shear due to live load (kN) VLONG. MOM. = shear force resulting from column longitudinal moment (kN) I-8

Vn = nominal shear resistance (kN) Vp = plastic shear capacity (kN) Vs = shear resistance provided by shear reinforcement (kN) VSERV. II = shear from Service II load combination (kN) Vs,MAX = maximum displacement corresponding to po (mm) Vsr = shear force range determined for the fatigue limit state (kN) VSTR. I = shear for Strength I load combination (kN) VT = shear due to torsion (kN) VTEQ = shear due to transverse earthquake load (kN) VTRANSV. MOM. = shear force resulting from column transverse moment (kN) Vu = factored shear force (kN) W = total nominal, unfactored dead load of the bridge superstructure and tributary substructure (N) w = width of compression flange between longitudinal stiffeners or distance from the web to the nearest longitudinal stiffener (mm); width of pier cap between web plates (m) Zr = shear fatigue strength of a shear connector (kN) α = offset factor β = factor indicating ability of diagonally cracked concrete to transmit tension βd = ratio of maximum factored permanent load moments to maximum factored total load moment, always positive ∆ = sidesway (mm) (∆F)n = nominal fatigue resistance (MPa) (∆F)TH = constant amplitude fatigue threshold (MPa) (∆f) = live load stress range due to the passage of the fatigue load (MPa) (∆M)FATIGUE = range in live load moment due to the passage of the fatigue load (kN-m) (∆ML)FATIGUE = range in longitudinal column live load moment due to the passage of the fatigue load (kN-m) (∆MT)FATIGUE = range in transverse column live load moment due to the passage of the fatigue load (kN-m) (∆P)FATIGUE = range in axial live load due to the passage of the fatigue load (kN) δb = moment magnifier for braced mode deflection δs = moment magnifier for unbraced mode deflection γ = load factor for the fatigue load combination γEQ = load factor for live load in Extreme Event Load Combination I γP = load factor for permanent loads ϕ = resistance factor ϕf = resistance factor for flexure ϕsc = resistance factor for shear connectors ϕu = resistance factor for fracture of tension members ϕy = resistance factor for yielding of tension members λb = coefficient related to b/t ratio θ = angle of inclination of diagonal compressive stresses (DEG) ρs = volumetric ratio of spiral reinforcing I3 GENERAL OVERVIEW The primary components of a bridge having girders integral with an intermediate pier are shown in Figures I-1 and I-2. The following is an overview of the design procedure for bridges having girders integral with intermediate piers: A. Develop general bridge dimensions (e.g., roadway width, span arrangements, girder spacing, and column height). B. Determine preliminary member sizes. C. Determine member forces for all applicable loads. D. Design column for controlling load combinations in accordance with current AASHTO LRFD Specifications and the proposed specifications herein. As discussed previously in the report, intermediate piers will typically consist of single column bents for which the proposed specifications apply. Multi-column bent applications will be rare, except in the case of outriggers; therefore, multi-column bents are not covered by the proposed specifi- cations or within the design example. However, as mentioned in the body of the report and in Appendices A I-9

through H, which are provided on the accompanying CD-ROM, the design process of the integral connection is applicable to multi-column piers and their integral connections. Once the analyses are completed, the main dif- ference between single-column piers and multi-column piers is that the top regions of the columns in multi- column piers are subjected to significant moments in both the longitudinal and transverse directions while these regions in single-column piers are essentially subjected to longitudinal moments. The column transverse moments may be transferred to the pier cap using the same procedure illustrated in this example for the longi- tudinal moment of the single-column pier. E. For bridges located within seismic regions, check preliminary girder sizes for forces from the Strength and Extreme Event I limit states. The design forces due to seismic loading shall be taken as the lesser of the forces from an elastic analysis divided by the applicable response modification factor or those associated with the plas- tic hinging of the column. F. Design cap beam in accordance with current box-beam design provisions in the AASHTO LRFD Specifications. For seismic loads, design for the lesser of the elastic forces divided by the applicable response modification fac- tor or those associated with the plastic hinging of the column. Notice that the cap beam is subjected to vertical and horizontal shear forces and the moments associated with them. It is also subjected to torsion. The magnitude of torsion transferred to the pier cap at each girder location is equal to the algebraic difference in girder moment at either face of the pier cap as shown in Figure I-3. The moment at the top of the column is equal to the sum of the torsional moments applied to the pier cap. Figure I-4 shows schematically the torsional moments on the pier cap and the column top moment. G. Design the girder-to-cap beam connection components shown in Figure I-5. I-10 30500 mm 30500 mm Pier Cap (100 ft) (100 ft) 7620 mm (25 ft) Single Column Girder Figure I-1. Bridge elevation. 11450 mm (37 ft - 6 in.) Out-to-Out 10950 mm (35 ft - 11 in.) Curb-to-Curb 3 spaces @ 3050 mm (10 ft) = 9150 mm (30 ft) 1830 mm (6 ft) 210 mm (8.5 in.) 1150 mm 1150 mm (3 ft - 9 in.)(3 ft - 9 in.) Pier CapGirderConnection to Pier Cap (Typ.) Figure I-2. Bridge cross section.

I-11 Pier Cap Theoretical Moment Diagram Girder Column Torsion transferred from girder to pier cap Girder Design Moments Torsion transferred to pier cap at girder location (Typ.) Longitudinal moment at top of column equal to sum of torsional moments Bolted double-angle connection for shear Flange splice plate for moment Intermediate diaphragm Figure I-3. Girder moments and pier cap torsion. Figure I-4. Pier cap torsion and column moment. Figure I-5. Girder-to-cap beam connection.

1. Design double-angle bolted connection for the maximum vertical shear in the girders. 2. Design flange splice plates for the maximum negative moment in the girders. The design force for the splice plates shall be taken equal to the maximum negative girder moment divided by the girder depth as shown in Figure I-6. 3. Design splice plate connection to cap beam. The design shear force for the connection shall be taken equal to the maximum torsion transferred to the pier cap divided by the depth of the pier cap as shown in Figure I-6. H. Design shear studs for column-to-cap beam connection (see Figure I-7). 1. Design shear studs located on bottom flange plate. Design these studs for the maximum horizontal shear at the top of the column (Figure I-7). 2. Design shear studs located on web plates of cap beam. These studs are designed for the shear produced from the longitudinal moment and axial load at the top of the column as shown in Figure I-8. 3. Design shear studs located on diaphragm plates. These studs shall be designed for the shear originating from the transverse moment at the top of the column as shown in Figure I-9. I4 DESIGN PARAMETERS The dimensions of the bridge are as shown in Figures I-1 and I-2. As shown in Figure I-1, the bridge is a two-span continuous structure with 30,500-mm (100-ft) span lengths. As shown in Figure I-2, the superstructure consists of a 210-mm (8.5-in.) thick cast-in-place reinforced concrete deck [10 mm (0.5 in.) of which is assumed nonstructural for an integral wearing surface in accordance with Article 2.5.2.4] supported by four steel I-girders spaced at 3,050 mm (10 ft). The girders are integral with a steel box-beam pier cap supported by a single reinforced concrete column. The column is 7,620 mm (25 ft) in height from the bottom of the pier cap to the top of the footing. Concrete for the deck and column was assumed to have a 28-day strength of 28 MPa (4 ksi). Unless noted other- wise, all structural steel was assumed to be ASTM A 709M (A 709), Grade 345W (50W). The bridge is located in Seismic Performance Zone 4 and will be designed for the greatest of either Extreme Event I or Strength I limit state forces. Only dead, live, and earthquake loads were considered. I5 PRELIMINARY DESIGN I5.1 I-Girders A conventional line girder design/analysis program was used to obtain preliminary plate sizes for the girders based on dead and live load requirements. The program does not take into account the effect of the integral connection on the girder loads. Initial plate sizes for both an interior and exterior girder were based on a design run for an interior girder of the bridge and are shown in Figure I-10. Final girder design will consist of a check on girder moments obtained I-12 V d PRT T V PLT MLT MRT Flange Splice Plate PRTPLT PRT = Flange splice plate force, right side = MRT / d T = Torsion on pier cap = MRT + MLT V = PRT + PLT = (MRT + MLT)/d = T/d MRTMLT PRT PRT PLT PLT Pier Cap Girder MRT = Girder moment, right side MLT = Girder moment, left side PLT = Flange splice plate force, left side = MLT / d V = Splice plate connection force Figure I-6. Splice plate and splice plate connection forces.

I-13 from a computer model that incorporates the stiffness of the pier cap and the rigid connection between the cap beam and girders. I5.2 Column An initial column diameter of 1,830 mm (6 ft) was selected based on past experience and preliminary design. I5.3 Pier Cap The dimensions of the pier cap, as shown in Figure I-11, were controlled by the size of the column and girders. The width of the pier cap is controlled by the diameter of the column and was set equal to the column diameter plus 225 mm (9 in) on each side of the column. The additional width of the cap is required to provide construction tolerance and to provide additional space for the cap beam to column connection steel. The height of the pier cap is set equal to the depth A A Section A-A Pier Cap Web Plate Girder Girder Column Pier Cap Diaphragm H = Column Horizontal Design Shear H Figure I-7. Shear studs for column-to-cap beam connection.

of the girders in the negative moment region. This enables the connection of a splice plate to the outside of the girder flanges and to the flange plate of the pier cap for the transfer of moment across the width of the pier cap. The initial plate thickness of the pier cap was selected to prevent buckling of the compression flange under maximum negative moment in accordance with Article 6.11.2.1.3a of the AASHTO LRFD Specifications. I6 COMPUTER MODEL Under live loading, the vertical movement of the pier cap and the rigid connection between the pier cap and the gird- ers will cause a bridge with an integral pier cap to differ from a conventional structure in three main areas. These areas are: the magnitude and location of maximum positive and negative bending moments in the girders, the maximum neg- ative bending moments and torsion in the pier cap, and the maximum moments in the pier. In addition, the ratio between the moments in the interior and exterior girders will be different compared to this ratio for a conventional structure. I-14 P = Column Axial Load V=P/2 ML = Longitudinal Column Moment V=ML/w w Shear stud forces due to column axial load Shear stud forces due to longitudinal column moment V=P/2 V=ML/w Figure I-8. Shear forces acting on shear studs located on web plates of cap beam. MT = Transverse Column Moment S V=MT/S V=MT/S Box-Beam Pier Cap Pier Cap Internal Diaphragm Girder Flange Splice Plate Figure I-9. Shear forces acting on shear studs located on internal diaphragms of pier cap.

For these reasons, a computer model of the structure was created for the purpose of determining the maximum live load forces in the components of the bridge. In addition, the model was also used to determine seismic forces within the bridge, although these forces could have been computed by hand. I6.1 Description of Model For the live load analysis of the bridge, a grillage model of the structure was created using the commercial software program SAP2000. In a grillage model, all parts of the bridge are modeled using beam elements. In particular, the struc- tural behavior of the concrete deck is included in the model by placing transverse beams between the main girders with flexural and torsional properties that mimic the action of the deck. This type of approach will give accurate results that are easy to interpret. Figure I-12 provides an overall view of the model. As shown in Figure I-12, roller supports allow- ing longitudinal movement but restraining transverse movement were assumed for the girders at the abutment locations and the column was assumed fixed at its base. Section properties for elements used in the model are given in Table I-1. I6.1.1 Girders Girders were modeled using beam elements positioned at the neutral axis location for the actual composite girder. Per Article 4.5.2.2 and the corresponding commentary, section properties for the girders were determined assuming I-15 21350 mm (70 ft) 18300 mm (60 ft) 21350 mm (70 ft) A A B B Section A-A Positive Moment Region 12 mm (0.5 in.) 25 mm (1 in.) 305 mm (12 in.) 1372 mm (54 in.) Section B-B Negative Moment Region 50 mm (2 in.) 380 mm (15 in.) 1372 mm (54 in.) 12 mm (0.5 in.) 12 mm(0.5 in.) 50 mm (2 in.) Figure I-10. Preliminary girder dimensions.

uncracked sections and that the girders were composite along their entire length. Since the model is being used for live load analysis, section properties correspond to the short-term composite section (n = 8). Both flexural and torsional properties were included. A haunch thickness equal to zero was assumed in calculating section properties. Where the girders frame into the pier cap, rigid links were used to account for the difference in neutral axis location between the two elements. Although the steel girders and pier cap have the same depth, the neutral axis of the girders is higher than that for the pier cap due to the composite action of the concrete deck. Thus, an end rotation of the gird- I-16 2280 mm (7 ft - 6 in.) 1472 mm (4 ft - 10 in.) 30 mm (1.25 in.) 30 mm (1.25 in.) Figure I-11. Preliminary cap beam dimensions. 11 sp a @ 2 500 (8 ' - 2 7/1 6") 11 sp a @ 2 500 (8 ' - 2 7/1 6") 11450 (37' - 6") See Figure H-13 for details Neg M om Regio n Pos M om Regio n 3 spa @ 3050 (10') = 9150 (30') xy z 1150 (3' - 9") 3050 0 (100 ') 3050 0 (100 ') 1650 (5' - 5" ) 850 (2 ' - 9 7/1 6") 2135 0 (70') 9150 (30') 2135 0 (70') 9150 (30') 3000 (9' - 1 0") 3000 (9' - 10 ") Transverse beams representing the deck slab Note: All dimensions are in mm unless noted otherwise. Figure I-12. SAP2000 schematic.

ers will introduce a small longitudinal displacement of the pier cap and the pier. The connection between the pier cap and girders is shown in Figure I-13. I6.1.2 Slab and Cross Frames The slab was modeled by beam elements spanning transversely across the width of the bridge. Each slab element models the effect of a 2,500 mm (8 ft − 27/16 in.) wide strip of concrete deck and is rigidly connected to the girder ele- ments. The moment of inertia for the slab elements is taken as the conventional second moment of area. The torsional constant, J, is given by the following equation: J = bt3/6 Where b = width of the deck represented by the element (mm) t = thickness of the deck (mm) The cross frames in the bridge, which occur at 0; 8,000; 15,500; and 23,000 mm (0 ft; 26 ft − 3 in.; 50 ft − 101/4 in.; 75 ft − 51/2 in.) along the length of each span starting from the abutment, were modeled by using equivalent beams. In determining the moment of inertia for the equivalent beams, the cross frames were analyzed as a truss. The moment of inertia for the equivalent beam was then taken as that which produced the same deflection as the truss when both I-17 Element Type Material A mm2 (in2) Ivert. x109 mm4 (in4) Ihoriz. x109 mm4 (in4) J x109 mm4 (in4) Shear Area mm2 (in2) Column Concrete 2,630,220 (4,077) 550.5 (1,322,582) 550.5 (1,322,582) 1,101 (2,645,163) ---- Int. Bm. – Pos. Mom. Steel 91,549 (141.9) 34.75 (83,487) 23.60 (56,699) 0.4888 (1,174) ---- Int. Bm. – Neg. Mom. Steel 119,264 (184.9) 36.65 (88,052) 42.70 (102,587) 0.5046 (1,212) ---- Ext. Bm. – Pos. Mom. Steel 88,349 (136.9) 30.11 (72,340) 23.34 (56,075) 0.4648 (1,117) ---- Ext. Bm. – Neg. Mom. Steel 115,664 (179.3) 31.22 (75,006) 42.15 (101,266) 0.4982 (1,197) ---- Pier Cap Steel 221,520 (343.4) 166.5 (400,018) 85.20 (204,694) 171.1 (411,069) 88,320 (136.9) Deck Concrete 500,000 (775.0) 260.4 (625,614) 1.670 (4,012) 3.330 (8,000) ---- Cross Frame Steel 68,120 (105.6) 260.4 (625,614) 2.760 (6,631) 0.4760 (1,144) ---- TABLE I-1 Element section properties for computer model 448 mm (1' - 5 5/8") Rigid Link (typ.) 736 mm (2' - 5") 7620 mm (25') Pier Cap Rigid End Zone Horizontal plane containing girders NA of pier cap NA of girders 448 mm (1' - 5 5/8") Figure I-13. Modeling details.

were subjected to the same loading conditions. The torsional constant for the equivalent beam was simply taken as that for the slab contributory to the element. I6.1.3 Pier Cap and Column The integral pier cap was modeled with beam elements. The pier cap is short and heavily loaded, therefore, deflec- tions due to shear will be significant and including the effect of shear deformation will improve the accuracy of the results. By entering a value for the shear area of the element, SAP2000 will consider shear deformations in the analy- sis. The shear area of the pier cap was taken equal to the area of the two side plates. The reinforced concrete column was modeled using beam elements. The column is assumed fixed at its base and is rigidly connected to the cap beam. A rigid end offset was used at the top of the column to account for the connection region where the column frames into the cap beam as shown in Figure I-13. The column stiffness was calculated based on an uncracked section. However, using a cracked section of the column to reflect the expected cracking under seis- mic loading is also acceptable. See Appendix D on the accompanying CD-ROM for further discussion regarding the use of cracked sections. I7 DEAD LOAD ANALYSIS Since the structure is symmetric under dead loads, the slope of the girders at the pier cap is zero and the column sees no moment. Since the moment in the column is zero, the flexural stiffness of the column and the torsional stiffness of the pier cap do not influence the behavior of the structure. However, the pier cap will deflect vertically and this will affect the distribution of forces between the girders and the magnitude and location of maximum design forces. Since the pier cap is relatively stiff, the deflection will be small and its effect on the behavior of the structure under dead load may be neglected. To confirm this conclusion, a study was performed to determine the influence of the pier cap deflection on member forces and support reactions. In the study, forces and reactions computed from a line girder analysis program assum- ing nonyielding supports were compared to the results obtained from a detailed computer model of the structure sim- ilar to that discussed in Section I6.1. Since the deflection of the pier cap is greatest at an exterior girder, the study was performed for an exterior girder only. The results of the study are presented in Tables I-2 and I-3 and pertain to dead loads applied to the noncomposite section. From Table I-2, one can see that the percent difference between the support reactions obtained from the two differ- ent types of analyses is small. In addition, except within a small region of the girder, the percent difference between dead load forces is also small, as shown in Table I-3. The location along the girder where the percent difference between dead load forces is high is were the magnitude of the force is small and therefore not critical to the design of the girder. There- fore, in analyzing the structure for dead loads, the influence of the deflection of the pier cap on member forces and reac- tions can be ignored and the structure can be treated as a conventional two-span structure with nonyielding supports. For this design example, a conventional line girder analysis program was used to determine dead load forces and reactions. Three separate runs were performed. These runs included: DC1 loads applied to the noncomposite steel sec- tion and DC2 and DW loads applied to the long-term composite (3n = 24) section. Dead loads used in the runs are given in Table I-4. DC1 loads consisted of the weight of the girder, miscellaneous steel, slab, and haunch. DC2 loads consisted of the weight of the parapet only. The future wearing surface (FWS) represents the DW load. For situations where the integral pier connection is not at the axis of symmetry (e.g., bridges with unequal spans) the pier cap will experience torsion and the column will be subjected to moment and shear for any dead loads applied after the joint region above the column has been poured. To determine the forces throughout the structure, a 2-dimensional frame analysis as indicated in the proposed specifications could be performed. Alternatively, a 3-dimensional refined analysis that takes into account the flexural stiffness of the column and the torsional stiffness of the pier cap could be used. I-18 Line Girder Analysis Program 3-D Computer Model Percent Difference Reaction at Abutment, kN (kips) 160.94 (36.18) 167.21 (37.59) +3.90% Reaction at Pier, kN (kips) 677.53 (152.32) 668.16 (150.21) -1.38% TABLE I-2 Noncomposite dead load reactions

For dead loads applied prior to pouring the joint region, there is no means by which forces can be transferred to the column. Since the column sees no moment, the effect of the flexural stiffness of the column and the torsional stiffness of the pier cap on the behavior of the structure can be ignored and the structure can be treated as a conventional struc- ture with nonyielding supports. I8 SEISMIC ANALYSIS I8.1 Seismic Design Parameters For this design example, assume the following parameters: Acceleration coefficient, A = 0.4 Importance category = essential Site coefficient, S = 1.2 (Soil Profile Type II) For an acceleration coefficient greater than 0.29, the bridge is assigned to Seismic Zone 4 in accordance with Table 3.10.4-1. I8.2 Method of Analysis Per Table 4.7.4.3.1-2, the bridge may be taken as regular since it is on tangent and the span length ratio from span to span is less than 3. From Table 4.7.4.3.1-1, for a multi-span, essential, regular bridge assigned to Seismic Zone 4, the minimum required method of analysis is the multi-mode spectral method. Typically, regular bridges will respond principally in their fundamental mode of vibration. Bridges that respond to earthquake ground motion in their fundamental modes of vibration can be analyzed by single mode methods such as I-19 Moment, kN-m (k-ft) Shear, kN (kips) Distance from Abut. m (ft) Girder Analysis Program Computer Model Percent Difference Girder Analysis Program Computer Model Percent Difference 0 (0) 0 (0) 0 (0) ---- 160.94 (36.18) 167.21 (37.59) 3.90% 3 (9.8) 411.51 (303.53) 430.31 (317.40) 4.57% 113.39 (25.49) 119.66 (26.90) 5.53% 5.5 (18.0) 645.47 (476.10) 679.93 (501.52) 5.34% 73.77 (16.58) 80.04 (17.99) 8.50% 8 (26.2) 780.36 (575.59) 830.49 (612.57) 6.42% 34.14 (7.67) 40.41 (9.08) 18.37% 10.5 (34.4) 816.19 (602.02) 861.07 (635.12) 5.50% -5.48 (-1.23) -7.58 (-1.70) 38.32% 13 (42.7) 752.96 (555.38) 792.58 (584.61) 5.26% -45.11 (-10.14) -47.21 (-10.61) 4.66% 15.5 (50.9) 590.66 (435.67) 625.03 (461.02) 5.82% -84.73 (-19.05) -86.83 (-19.52) 2.48% 18 (59.1) 329.31 (242.90) 340.90 (251.45) 3.52% -124.36 (-27.96) -133.46 (-30.00) 7.32% 20.5 (67.3) -31.11 (-22.95) -42.29 (-31.19) 35.94% -163.98 (-36.87) -173.09 (-38.91) 5.56% 21.35 (70.05) -176.22 (-129.98) -195.14 (-143.94) 10.74% -177.45 (-39.89) -186.56 (-41.94) 5.13% 23 (75.5) -493.02 (-363.65) -526.97 (-388.69) 6.89% -206.54 (-46.43) -215.65 (-48.48) 4.41% 25.5 (83.7) -1064.47 (-785.15) -1086.71 (-801.56) 2.09% -250.62 (-56.34) -245.93 (-55.29) 1.87% 28 (91.9) -1746.10 (-1287.92) -1756.64 (-1295.70) 0.60% -294.69 (-66.25) -290.01 (-65.20) 1.59% 30.5 (100.1) -2537.93 (-1871.97) -2536.76 (-1871.11) 0.05% -338.77 (-76.16) -334.08 (-75.11) 1.38% TABLE I-3 Noncomposite dead load forces

DC1 DC2 DW Girder Misc. Steela Slab Haunch b Parapet FWSc Int. Bm. – Pos. Mom. 2.14 (0.15) 0.21 (0.01) 15.08 (1.03) 0.27 (0.02) 4.74 (0.32) 4.49 (0.31) Int. Bm. – Neg. Mom. 4.19 (0.29) 0.21 (0.01) 15.08 (1.03) 0.00 (0.00) 4.74 (0.32) 4.49 (0.31) Ext. Bm. – Pos. Mom. 2.14 (0.15) 0.21 (0.01) 13.23 (0.91) 0.27 (0.02) 4.74 (0.32) 3.18 (0.22) Ext. Bm. – Neg. Mom. 4.19 (0.29) 0.21 (0.01) 13.23 (0.91) 0.00 (0.00) 4.74 (0.32) 3.18 (0.22) a Taken as 10% of the weight of the interior beam in the positive moment region. b Taken as 38 mm (1.5 in.) in the positive moment region. c Taken equal to 150 kg/2 (0.030 ksf) between curbs. TABLE I-4 Uniform dead loads, N/mm (kip/ft) the single-mode spectral method or the uniform load method. Since this bridge is highly regular and quite simple geo- metrically, it will respond predominantly in its fundamental mode of vibration. Therefore, reasonably accurate seis- mic forces can be obtained from single mode methods of analysis. For this example, seismic analysis was performed using the uniform load method of Article 4.7.4.3.2c. I8.3 Equivalent Transverse Static Earthquake Loading Calculate the transverse lateral stiffness, K, of the bridge. Eq. C4.7.4.3.2c-1 From the SAP model, the maximum transverse displacement, Vs,MAX, resulting from a uniform load of 1 N/mm (0.068 kip/ft) is 0.154 mm (0.006 in.). Calculate the total weight, W. The total weight consists of the superstructure, pier cap, and half of the column. Half of the column is assumed tributary to the superstructure and the other half is assumed tributary to the foundation. W = 2(42,700)22.44 + 2(42,700)20.59 + 2(18,300)24.22 + 2(18,300)22.37 + 19.54(9,150) + 61.93(7,620/2) + 61.93(1,412) The quantity 61.93(1,412) accounts for the weight of the concrete in the pier cap above the column. W = 5,882,145 N (1,322 kips) Eq. C4.7.4.3.2c-3 Eq. 3.10.6.1-1 2.5A = 2.5(0.4) = 1.0 C 1.2AST Asm m2 / 3 = ≤ 2 5. T sm = = 2 31 623 5 882 145 9 81 396 104 0 244 π . , , . ( , ) . T WgKm = 2 31 623 π . K 1(61, 000)0.154 396,104 N/mm (27,130 kip/ft)= = K p LV o s,MAX = I-20

Eq. C4.7.4.3.2c-4 I8.4 Equivalent Longitudinal Static Earthquake Loading Calculate the longitudinal stiffness, K, of the bridge. Eq. C4.7.4.3.2c-1 From the SAP model, the maximum longitudinal displacement, Vs,MAX, resulting from a uniform load of 1 N/mm (0.068 kip/ft) is 0.430 mm (0.017 in.). W = 5,882,145 N (1,322 kips) (from previous calculations in Section I8.3) Eq. C4.7.4.3.2c-3 Eq. 3.10.6.1-1 2.5A = 2.5(0.4) = 1.0 Eq. C4.7.4.3.2c-4 I8.5 Intermediate Pier Column Seismic Forces The equivalent uniform loads were applied to the computer model and a second static analysis was performed to obtain the seismic forces shown in Table I-5. I8.6 Evaluate Slenderness Effects For unbraced frames, per Article 5.7.4.3, slenderness effects may be neglected if the slenderness ratio, K
u/r, is less than 22. In computing the slenderness ratio, the unbraced length,
u, was taken as the distance from the top of p 1.0(5,882,145)61, 000 N/mm (6.60 kip/ft)e = = 96 43. p C WLe sm = C 1.2(0.4)1.2(0.408) Csm 2/3 sm= = > ∴ =1 047 1 0 1 0. . . C 1.2AST Asm m2 / 3 = ≤ 2 5. T sm = = 2 31 623 5 882 145 9 81 141 860 0 408 π . , , . ( , ) . T WgKm = 2 31 623 π . K 1(61, 000)0.430 N/mm (9,716 kip/ft)= = 141 860, K p LV o s,MAX = p 1.0(5,882,145)61, 000 N/mm (6.60 kip/ft)e = = 96 43. p C WLe sm = C 1.2(0.4)1.2(0.244) Csm 2/3 sm= = > ∴ =1 475 1 0 1 0. . . I-21

the foundation to the center of the pier cap. Arguably, this distance could have been taken as the distance from the top of the foundation to the bottom of the pier cap. For the effective length factor, K, use the recommended design values from Table C4.6.2.5-1. As specified in Article C5.7.4.3, r is taken as 0.25 times the diameter of the column. Find the slenderness ratio of the column in the transverse direction. Under the equivalent transverse static earthquake load, the column of a single column pier acts as a cantilever that resembles Case (e) from Table C4.6.2.5-1, thus, take K equal to 2.1. The cantilever-type behavior may also be con- firmed by checking the deflected shape of the column in the computer output. Find the slenderness ratio of the column in the longitudinal direction. The rotation of the column is restrained by the connection to the superstructure at the top and the footing at the bot- tom. This allows the behavior of the column to be represented by Case (c) from Table C4.6.2.5-1, thus, take K equal to 1.2. This conclusion may be confirmed by checking the deflected shape of the column in the computer output. I8.7 Moment Magnification Per Article 5.7.4.3, for compression members with a slenderness ratio less than 100, the effects of slenderness may be approximated by the moment magnification method, as specified in Article 4.5.3.2.2b. As specified in Article 3.10.8, seismic forces from the two static earthquake loadings are combined to form two load cases for design. Load Case 1 is taken equal to 100 percent of the force effects from the longitudinal earthquake load plus 30 percent of the force effects from the transverse earthquake load. Load Case 2 is taken equal to 100 percent of the force effects from the transverse earthquake load plus 30 percent of the force effects from the longitudinal earth- quake load. For the moments shown in Table I-5, one can see that Load Case 1 will control the design of the column. For circular columns, the longitudinal reinforcing is designed for the moment Mu computed as follows: Where ML = Moment in longitudinal direction (kN-m) MT = Moment in transverse direction (kN-m) Since 30 percent of either transverse moment shown in Table I-5 is considerably less than the corresponding longi- tudinal moment, any increase in MT due to slenderness effects will have a negligible effect on the magnitude of the design moment. Therefore, moment magnification to account for slenderness effects in the transverse direction can be ignored. However, for completeness, calculations for the magnification of the transverse moment have been incorpo- rated within this design example. Determine the magnified factored moments in the transverse direction. M M Mu L 2 T 2= ( ) + ( ) Kl r 1.2(8,356) 0.25(1,830) neglect slenderness effects u = = < ∴21 9 22. Kl r 2.1(8,356) 0.25(1,830) consider the effects of slenderness u = = > ∴38 4 22. I-22 Longitudinal Earthquake Load Transverse Earthquake Load Longitudinal Direction Transverse Direction Longitudinal Direction Transverse Direction Column Location Axial kN (kips) Shear kN (kips) Mom. kN-m (k-ft) Shear kN (kips) Mom. kN-m (k-ft) Axial kN (kips) Shear kN (kips) Mom. kN-m (k-ft) Shear kN (kips) Mom. kN-m (k-ft) Top 0 (0) 5883 (1323) 12932 (9539) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 948 (213) 904 (667) Bottom 0 (0) 5883 (1323) 31895 (23526) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 948 (213) 8125 (5993) TABLE I-5 Column seismic forces

Mc = δbM2b + δsM2s Eq. 4.5.3.2.2b-1 M2s is defined as the moment due to factored lateral loads that result in sidesway, ∆, greater than
u/1,500. From the computer model, for the equivalent transverse static earthquake load, ∆ = 13.7 mm (0.54 in.).
u/1500 = 8,356/1,500 = 5.6 mm (0.22 in.) ∆ >
u/1,500, therefore, M2b = 0 (M2s)TOP = 904 kN-m (667 k-ft) (M2s)BOTT = 8,125 kN-m (5,993 k-ft) Eq. 4.5.3.2.2b-4 Eq. 4.5.3.2.2b-5 Since the column reinforcing is not known at this point, use Equation 5.7.4.3-2 to determine EI. Eq. 5.7.4.3-2 βd = 0 Ig = 5.505 × 1011 mm4 (1,322,582 in.4) Ig = 5.505x1011 mm4 (1,322,582 in4) Use the maximum load factors for dead load in determining the factored axial load since s is maximized when Pu is maximized. ΣPu = 1.25PDC + 1.5PDW ΣPu = 1.25(3,880) + 1.5(608) = 5,762 kN (1,295 kips) see Section I9.1 for PDC & PDW δs = − = 1 1 5 7620 5 179 270 1 07 , . ( , ) . P 1, 000 2.1(8,356) kN (40,304 kips)e 2 = ×( ) [ ] = π 5 593 10 179 270 15 2 . , EI N-mm 1.949 10 k-in2 9 2= ×( ) + = × ×( ) 25 399 5 505 0 2 5 1 0 5 593 10 11 15 , . . . E f MPa (3,684 ksi)c = ′ = =4 800 4 800 28 25 399, , ,c EI E Ic g d = + 2 5 1 . β P EI Ke 2 u 2= ( ) π l δ ϕ s u e P P = − ∑ ∑ 1 1 I-23

(Mc)TOP = 1.07(904) = 967 kN-m (713 k-ft) (Mc)BOTT = 1.07(8,125) = 8,694 kN-m (6,413 k-ft) Column moments adjusted for the effects of slenderness are given in Table I-6. I9 COLUMN DESIGN I9.1 Column Dead Load Forces Since the bridge is symmetric, the column experiences only axial compression due to dead load. Axial compression in the column is computed from the loading diagrams shown in Figure I-14. Beam reactions applied to the cap beam were taken from a conventional line girder analysis program. (PDC)TOP = 2(941) + 2(866) + 9.15(19.54) + 1.412(61.93) = 3,880 kN (872 kips) The quantity 1.412(61.93) accounts for the weight of the concrete in the cap beam above the column. (PDC)BOTT = 3,880 + 7.62(61.93) = 4,352 kN (978 kips) PDW = 2(126) + 2(178) = 608 kN (137 kips) I9.2 Column Live Load Forces Column live load forces were obtained from the computer model and are shown in Table I-7 for three live load cases. The live load cases are as follows: Live Load Case 1—maximum transverse moment with corresponding longitudinal moment and axial load. Live Load Case 2—maximum longitudinal moment with corresponding transverse moment and axial load. Live Load Case 3—maximum axial load with corresponding longitudinal and transverse moments. I9.3 Load Cases for Design Applicable load combinations: Extreme Event I − γPDC + γPDW + γEQLL + 1.0EQ Strength I − γPDC + γPDW + 1.75LL Where γP = Load factor for permanent loads as prescribed in Table 3.4.1-2. γEQ = Load factor for live load in Extreme Event Load Combination I. As discussed in the commentary to Article 3.4.1, an appropriate value for this factor is still unresolved. Based upon past editions of the Standard Speci- fications, a value of 0.0 was used for this example. I-24 Longitudinal Earthquake Load Transverse Earthquake Load Column Location Long. Moment kN-m (k-ft) Transv. Moment kN-m (k-ft) Long. Moment kN-m (k-ft) Transv. Moment kN-m (k-ft) Top 12932 (9539) 0 (0) 0 (0) 967 (713) Bottom 31895 (23526) 0 (0) 0 (0) 8694 (6413) TABLE I-6 Magnified column moments for earthquake load

I9.3.1 Load Cases for Extreme Event I Load Combination Per Article 3.10.9.4.1, design column for the modified design moment was determined in accordance with Arti- cle 3.10.9.4.2. MMOD. = Mu / R From Table 3.10.7.1-1, for a single column of an essential structure, R = 2.0. I-25 19.54 N/mm (1.34 kip/ft) 866 kN (195 kips) 941 kN (212 kips) 61.93 N/mm (4.24 kip/ft) a) Unfactored DC Load 126 kN (28 kips) 178 kN (40 kips) b) Unfactored DW Load 1525 mm (5 ft) 714 mm (2' - 4 1/16")811 mm (2' - 7 15/16") 3050 mm (10 ft) Face of column pier cap pier cap girder reactions (TYP.) girder reactions (TYP.) 4575 mm (15 ft) 941 kN (212 kips) 866 kN (195 kips) 126 kN (28 kips) 178 kN (40 kips) Figure I-14. Column dead loads.

For circular columns, As discussed in Section I8.7, the load combination pertaining to 100 percent of the force effects from the longitu- dinal earthquake load (LEQ) plus 30 percent of the force effects from the transverse earthquake load (TEQ) will con- trol the design of the column. Therefore, ML = 1.0(ML)LEQ + 0.3(ML)TEQ MT = 1.0(MT)LEQ + 0.3(MT)TEQ For the values given in Table I-6, the design moments including the response modification factor at the top and bot- tom of the column are as follows: (ML)TOP = 1.0(12,932) + 0.3(0.0) = 12,932 kN-m (9,539 k-ft) (MT)TOP = 1.0(0.0) + 0.3(967) = 290 kN-m (214 k-ft) (MMOD.)TOP = 12,935/R = 12,935/2 = 6,468 kN-m (4,771 k-ft) (ML)BOTT. = 1.0(31,895) + 0.3(0.0) = 31,895 kN-m (23,526 k-ft) (MT)BOTT. = 1.0(0.0) + 0.3(8,694) = 2,608 kN-m (1,924 k-ft) (MMOD.)BOTT. = 32,001/R = 32,001/2 = 16,001 kN-m (11,802 k-ft) Factored design forces for the load cases of Load Combination Extreme Event I are given in Table I-8. M 31,895 2,608 kN-m (23,604 k-ft)u BOTT. 2 2( ) = ( ) + ( ) = 32 001, M 12,932 290 kN-m (9,541 k-ft)u TOP 2 2( ) = ( ) + ( ) = 12 935, M M Mu L 2 T 2= ( ) + ( ) I-26 Column Location P kN (kips) MT kN-m (k-ft) ML kN-m (k-ft) Top 1932 (434) 3417 (2520) 19 (14) Live Load Case 1 Bottom 1932 (434) 438 (323) 19 (14) Top 1218 (274) 307 (226) 2489 (1836) Live Load Case 2 Bottom 1218 (274) 39 (29) 2489 (1836) Top 2465 (554) 623 (460) 25 (18) Live Load Case 3 Bottom 2465 (554) 80 (59) 25 (18) TABLE I-7 Unfactored column live load forces Column Location Load Case DC load factor PDC kN (kips) DW load factor PDW kN (kips) PEQ kN (kips) Pu kN (kips) MMOD. kN-m (k-ft) LC1 1.25 3880 (872) 1.5 608 (137) 0 (0) 5762 (1295) 6468 (4771) Top LC2 0.9 3880 (872) 0.65 608 (137) 0 (0) 3887 (874) 6468 (4771) LC1 1.25 4352 (978) 1.5 608 (137) 0 (0) 6352 (1428) 16001 (11802) Bottom LC2 0.9 4352 (978) 0.65 608 (137) 0 (0) 4312 (969) 16001 (11802) TABLE I-8 Load cases for extreme Event I load combination

I9.3.2 Load Cases for Strength I Load Combination For the moments in Table I-7, one can see that the design of the column for the Strength I Limit State will be con- trolled by the moment at the top of the column. For the longitudinal and transverse moments given in Table I-7, fac- tored live load design moments at the top of the column are as follows: Factored design forces at the top of the column for the load cases of Load Combination Strength I are given in Table I-9. I9.4 Longitudinal Reinforcing I9.4.1 Controlling Load Case Typically, load cases for the design of columns on highway bridges fall below the balance point on the interaction diagram for the column. Investigating the load cases for the Strength I load combination in Table I-9, one can see that Load Cases 1a and 2a will control for the Strength I limit state. Comparing Load Cases 1a and 2a to those for the Extreme Event I load combination in Table I-8 and considering that the resistance factor, ϕ, for the extreme event limit state is less than that for the strength limit state (ϕ = 0.5 compared to 0.75 for the strength limit state), one can see that the extreme event limit state will control the design of the column. It can also be seen that the bottom of the column will control the design. Size column based on the design forces at the bottom of the column. Limit maximum bar size to #36 (#11 USCU) to keep development lengths reasonable. Try a 1,830 mm (6 ft) diameter column with sixty-eight #36 (#11 USCU) bars bundled in groups of two for a total of thirty-four bundles. The interaction diagram for the factored resistance at the bottom of the column is plotted in Figure I-15. From the figure, one can see that the load cases given in Table I-8 for the bottom of the column plot inside the interaction diagram, therefore, okay. Terminate one bar in each bundle around two-thirds the column height leaving thirty-four #36 (#11 USCU) bars for strength at the top of the column. The interaction diagram for the factored resistance at the top of the column is plot- ted in Figure I-16. From the figure, one can see that the load cases given in Table I-8 for the top of the column plot inside the interaction diagram, therefore, okay. M ( 5 ( 23 kN-m (805 k-ft)LL LC3 2 2( ) = + =1 75 2 6 1 091. ) ) , M ( , 489 ( 07 kN-m (3,237 k-ft)LL LC2 2 2( ) = + =1 75 2 3 4 389. ) ) , M ( ( kN-m (4, 411 k-ft)LL LC1 2 2( ) = + =1 75 19 3 417 5 980. ) , ) , M M MLL L 2 T 2= ( ) + ( )1 75. I-27 Load Case DC load factor PDC kN (kips) DW load factor PDW kN (kips) LL load factor PLL kN (kips) Pu kN (kips) Mu kN-m (k-ft) LC1 1.25 3880 (872) 1.5 608 (137) 1.75 1932 (434) 9143 (2056) 5980 (4411) LC1a 0.9 3880 (872) 0.65 608 (137) 1.75 1932 (434) 7268 (1634) 5980 (4411) LC2 1.25 3880 (872) 1.5 608 (137) 1.75 1218 (274) 7894 (1775) 4389 (3237) LC2a 0.9 3880 (872) 0.65 608 (137) 1.75 1218 (274) 6019 (1353) 4389 (3237) LC3 1.25 3880 (872) 1.5 608 (137) 1.75 2465 (554) 10076 (2265) 1091 (805) LC3a 0.9 3880 (872) 0.65 608 (137) 1.75 2465 (554) 8201 (1844) 1091 (805) TABLE I-9 Load cases for Strength I load combination

I9.4.2 Development Length Check that the required development length for a straight #36 (#11 USCU) bar can be accommodated within the joint region atop the column shown in Figure I-17. As specified in Article 5.11.2.1.1, the tension development length,
d, shall not be less than the basic tension devel- opment length,
db, modified by any applicable modification factors that increase or decrease
db. For #36 (#11 USCU) bar and smaller, but not less than 0.06dbfy = 0.06(35.8)420 = 902 mm (35.51 in.) Modify the basic development length by 0.75 for reinforcement enclosed within a spiral composed of bars of not less than 6 mm (0.25 in.) in diameter and spaced at not more than a 100 mm (4 in.) pitch per Article 5.11.2.1.3. In addition, increase the basic development length by 1.25 to account for the overstrength capacity of the reinforcing, as specified in Article 5.10.11.4.3.
d = 0.75(1.25)
db = 0.75(1.25)1,597 = 1,497 mm (58.94 in.) ldb b y c A f f ( 28 mm (62.87 in.)= ′ = = 0 02 0 02 1 006 420 1 597. . , ) , controls I-28 0 5000 10000 15000 20000 25000 30000 35000 40000 0 5000 10000 15000 20000 ϕMn (kN-m) ϕP n (k N ) LC1 LC2 68 #36 (#11 USCU) Bars AS/Ag = 2.6% ϕ = See Article 5.10.11.4.1b Figure I-15. Interaction diagram for factored resistance at bottom of column. 0 5000 10000 15000 20000 25000 30000 35000 0 2000 4000 6000 8000 10000 12000 ϕMn (kN-m) ϕP n (k N ) LC1 LC2 34 #36 (#11 USCU) Bars AS/Ag = 1.3% ϕ = See Article 5.10.11.4.1b Figure I-16. Interaction diagram for factored resistance at top of column.

Length available within pier cap (see Figure I-11).
available = 1,472 − 60 = 1,412 mm (55.59 in.) < 1,497 mm (58.94 in.) ∴ No Good Extend bars through the top of the pier cap and into the deck 85 mm (3.35 in.).
available = 1,412 + 85 = 1,497 mm (58.94 in.) = 1,497 mm (58.94 in.) ∴ OK The design of the column-to-footing connection is not addressed in this example; therefore, a check on the devel- opment length for the bundled bars at the bottom of the column is not presented I9.5 Column Overstrength Determine column overstrength moments in accordance with Article 3.10.9.4.3b. For reinforced concrete columns, MOVRSTR. = 1.3Mn Interaction diagrams for the nominal resistance at the top and bottom of the column, for the assumed column diam- eter and reinforcing, are plotted in Figures I-18 and I-19. From Figure I-18, the nominal moment resistance at the top of the column for the axial loads given in Table I-8, with Pu = Pn is as follows: I-29 Column Reinforcement Spiral Reinforcement Reinforced Concrete Deck Box-Beam Pier Cap Bridge Girder Flange Splice Plate Height available for development of long. reinf. (can extend into deck for additional development length if necessary) Column Figure I-17. Development length for column longitudinal reinforcement.

LC1: (Mn)TOP = 13,218 kN-m (9,750 k-ft) LC2: (Mn)TOP = 12,392 kN-m (9,140 k-ft) From Figure I-19, the nominal moment resistance at the bottom of the column for the axial loads given in Table I-8 with Pu − Pn is as follows: LC1: (Mn)BOTT. = 21,406 kN-m (15,789 k-ft) LC2: (Mn)BOTT. = 20,712 kN-m (15,277 k-ft) Calculate the column overstrength moment at the top of the column. LC1: (MOVRSTR.)TOP = 1.3(13,218) = 17,184 kN-m (12,675 k-ft) LC2: (MOVRSTR.)TOP = 1.3(12,392) = 16,109 kN-m (11,882 k-ft) Calculate the column overstrength moment at the bottom of the column. I-30 0 10000 20000 30000 40000 50000 60000 70000 80000 0 5000 10000 15000 20000 ϕMn (kN-m) ϕP n (kN ) 34 #36 (#11 USCU) bars AS/Ag = 1.3% ϕ = 1.0 (ϕPn, ϕMn) (5,762 kN, 13,218 kN-m) (ϕPn, ϕMn) (3,887 kN, 12,392 kN-m) Figure I-18. Interaction diagram for nominal resistance at top of column. 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 0 5000 10000 15000 20000 25000 30000 ϕMn (kN-m) ϕP n (k N ) (ϕPn, ϕMn) (6,352 kN, 21,406 kN-m) 68 #36 (#11 USCU) bars AS/Ag = 2.6% ϕ = 1.0 (ϕPn, ϕMn) (4,312 kN, 20,712 kN-m) Figure I-19. Interaction diagram for nominal resistance at bottom of column.

LC1: (MOVRSTR.)BOTT. = 1.3(21,406) = 27,828 kN-m (20,526 k-ft) LC2: (MOVRSTR.)BOTT. = 1.3(20,712) = 26,925 kN-m (19,860 k-ft) I9.6 Spiral Reinforcing I.9.6.1 Design Shear Force Determine the design shear force for the column in accordance with Articles 3.10.9.4.1 and 3.10.9.4.3d. For the free body diagram of the column shown in Figure I-20, the shear force V equals the sum of the moments at the top and bottom of the column divided by the column height. V = (MTOP + MBOTT.)/h The moment at the top and bottom of the column is taken as the lesser of the design moment including a response modification factor of 1.0, rather than the value of 2.0 used to determine M or the moment associated with plastic hing- ing of the column (MOVRSTR.). Since the shear resistance of the concrete decreases for axial loads less than 0.1f ′cAg in the end regions of the column per Article 5.10.11.4.1c, shear forces associated with both the minimum and maximum axial loads on the column need to be investigated. Shear force for Load Case 1 (maximum axial load) MTOP = lesser of (MMOD.)TOP or (MOVRSTR.)TOP (MMOD.)TOP = Mu/R = 12,935/1.0 = 12,935 kN-m (9,541 k-ft) see Section I9.3.1 for Mu (MOVRSTR.)TOP = 17,184 kN-m (12,675 k-ft) (see Section I9.5) MTOP = lesser of 12,935 and 17,184 = 12,935 kN-m (9,541 k-ft) MBOTT. = lesser of (MMOD.)BOTT. or (MOVRSTR.)BOTT. (MMOD.)BOTT. = Mu/R = 32,001/1.0 = 32,001 kN-m (23,604 k-ft) see Section I9.3.1 for Mu I-31 PTOP V V h PBOTT MBOTT MTOP V = MTOP + MBOTT h Figure I-20. Free body diagram of column.

(MOVRSTR.)BOTT. = 27,828 kN-m (20,526 k-ft) (see Section I9.5) MBOTT. = lesser of 32,001 and 27,828 = 27,828 kN-m (20,526 k-ft) (Vu)LC1 = (12,935 + 27,828)/7.62 = 5,349 kN (1,203 kips) Since the shear resistance of the concrete is less for smaller axial loads, take the corresponding axial load in the column for shear design as that at the top of the column (as opposed to the axial load at the bottom of the column). Therefore, (Pu)LC1 = 5,762 kN (1,295 kips) (see Table I-8) Shear force for Load Case 2 (minimum axial load) MTOP = lesser of (MMOD.)TOP or (MOVRSTR.)TOP (MMOD.)TOP = Mu / R = 12,935/1.0 = 12,935 kN-m (9,541 k-ft) see Section I9.3.1 for Mu (MOVRSTR.)TOP = 16,109 kN-m (11,882 k-ft) (see Section I9.5) MTOP = lesser of 12,935 and 16,109 = 12,935 kN-m (9,541 k-ft) MBOTT. = lesser of (MMOD.)BOTT. or (MOVRSTR.)BOTT. (MMOD.)BOTT. = Mu / R = 32,001/1.0 = 32,001 kN-m (23,604 k-ft) see Section I9.3.1 for Mu (MOVRSTR.)BOTT. = 26,925 kN-m (19,860 k-ft) (see Section I9.5) MBOTT. = lesser of 32,001 and 26,925 = 26,925 kN-m (19,860 k-ft) (Vu)LC2 = (12,935 + 26,925)/7.62 = 5,231 kN (1,176 kips) (Pu)LC2 = 3,887 kN (874 kips) (see Table I-8) I.9.6.2 Shear Resistance of the Concrete Eq. 5.8.3.3-3 Per Article C5.8.2.9, dv can be taken as 0.9de, where de = D/2 + Dr/π Eq. C5.8.2.9-2 Where D = diameter of the column (mm) Dr = diameter of the circle passing through the centers of the longitudinal reinforcement (mm) de = 1,830/2 + [1,830-2(50)-36]/π = 1,454 mm (57.24 in.) dv = 0.9de = 0.9(1,454) = 1,309 mm (51.54 in.) Per the simplified procedure of Article 5.8.3.4.1, take β equal to 2.0. V f b dc c v v= ′0 083. β I-32

In end regions, for axial loads below 0.1f ′cAg, reduce shear resistance in accordance with Article 5.10.11.4.1c. 0.1f ′cAg = 0.1(28)π(1,830)2/[4(1,000)] = 7,365 kN (1,656 kips) End region shear resistance for Load Case 1. (Vc)LC1 = Vc (Pu)LC1/(0.1f ′cAg) = 2,104(5,762)/7,365 = 1,646 kN (370 kips) End region shear resistance for Load Case 2. (Vc)LC2 = Vc (Pu)LC2/(0.1f ′cAg) = 2,104(3,887)/7,365 = 1,110 kN (250 kips) I.9.6.3 Spacing of Spiral Reinforcing An equation giving the required spacing for the spiral reinforcing can be obtained as follows: ϕVn > Vu Vn = Vc + Vs Eq. 5.8.3.3-1 Substituting Equation 5.8.3.3-1 for Vn and rearranging gives ϕVs > Vu − ϕVc Vs = Avfydvcotθ/s Eq. C5.8.3.3-1 Substituting Equation C5.8.3.3-1 for Vs and solving for s gives s < ϕAvfydvcotθ/(Vu − ϕVc) Per the simplified procedure of Article 5.8.3.4.1, take θ equal to 45 degrees. For the Extreme Event limit state, ϕ = 1.0 per Article 1.3.2.1. Determine the required spacing of the spiral reinforcing within the end regions of the column. Load Case 1: Vu − ϕVc = 5,349-1.0(1,646) = 3,703 kN (833 kips) Load Case 2: Vu − ϕVc = 5,231-1.0(1,110) = 4,121 kN (926 kips) ← controls Try #16 (#5 USCU) bar for spiral reinforcing, Av = 400 mm2 (0.62 in2). s < 1.0(400)420(1,309)cot 45°/[4,121(1,000)] s < 53 mm (2.09 in.) → say s = 50 mm (2 in.) Check clear spacing between bars of spiral. clr. spacing = 50 − 16 = 34 mm (1.34 in.) > 25 mm (1 in.) per Article 5.10.6.2 ∴ OK Use #16 (#5 USCU) bar for spiral with spacing not to exceed 53 mm (2.09 in.) within the end regions of the column. Determine required spacing of spiral reinforcing between the end regions of the column. V kN (473 kips)c = =0 083 2 0 28 1 830 1 3091 000 2 104 . ( . ) ( , ) , , , I-33

Between the end regions of the column the shear resistance of the concrete is not reduced for axial loads less than 0.1f ′cAg; therefore, Load Case 1 will control the spacing of the spiral reinforcing. Vu − ϕVc = 5349-1.0(2104) = 3,245 kN (730 kips) s < 1.0(400)420(1,309)cot 45°/[3,245(1,000)] s < 68 mm (2.68 in.) Check the volumetric ratio of the spiral reinforcing as specified in Article 5.10.11.4.1d. ρs > 0.12f ′c /fy Eq. 5.10.11.4.1d-1 Where, Asp = cross-sectional area of spiral reinforcing (mm2) Asp = 200 mm2 (0.31 in2) for #16 (#5 USCU) bar dc = outside diameter of spiral (mm) dc = 1,830-2(50)+2(16) = 1,762 mm (69.37 in.) The critical value for the volumetric ratio occurs for the spacing between the end regions of the column. 0.12f ′c / fy = 0.12(28)/420 = 0.008 (ρs)provided < 0.12f ′c /fy, therefore, decrease spiral pitch between the end regions of the column. Find the spacing of the spiral reinforcing that satisfies Equation 5.10.11.4.1d-1. Solving for s gives s < 57 mm (2.24 in.) → say 50 mm (2 in.) In addition, provide spiral reinforcing within the center compartment of the pier cap according to the proposed design specifications. Within the connection region, the required spiral reinforcement shall be greater than half the amount provided in the plastic hinge region. I9.7 Column Reinforcing Details Refer to Figure I-21 for final column reinforcing details. s 4(200)420 0.12(28)1, 762≤ s 4A f 0.12f d sp y c c ≤ ′ 4A d s f f sp c c y≥ ′0 12. / ρs provided 4 200 1, 762 68( ) = ( ) ( ) = 0 0067. ρs provided sp c 4A d s( ) = I-34

I-35 Design of column to footing connection is not addressed in this design example B B AA 85 mm (3.35 in.) Section B-B #16 (#5 USCU) spiral reinforcement 34 - #36 (#11 USCU) bars Section A-A #16 (#5 USCU) spiral reinforcement 68 - #36 (#11 USCU) barsbundled in groups of two for 34 bundles. Terminate one bar in each bundle outside column end region where no longer needed for flexure. (Not all bundles are shown.) 76 20 m m (2 5 f t) S 5 3 m m (2 .09 in . ) En d R eg io n = 18 30 m m (6 ft) se e Ar tic le 5 .1 0. 11 .4 .1 c En d R eg io n = 18 30 m m (6 ft) se e Ar tic le 5 .1 0. 11 .4 .1 c 39 60 m m (1 3 f t) S 57 m m (2 . 24 in .) Extend longitudinal reinforcing through cap beam and 85 mm (3.35 in.) into deck. #16 (#5 USCU) spiral reinforcement S 100 mm (4 in.) Close spiral with 2 turns Close spiral with two turns and terminate with a seismic hook in accordance with Article 5.10.2.2. #16 (#5 USCU) spiral reinforcement W ith in En d Re gi on S 5 3 m m (2 .09 in . ) W ith in En d Re gi on ≤ Sa y S = 50 m m (2 in . ) Sa y S = 50 m m (2 in . ) Sa y S = 50 m m (2 in .) Figure I-21. Column reinforcement details.

I10 BEAM DESIGN The girders are rigidly connected to the pier cap, therefore, the column top moment is transferred to the girders. I10.1 Earthquake Loading Check preliminary girder size for moments developed during a seismic event. Design girders for the lesser of the elastic moments computed in accordance with Article 3.10.8 or the moments associated with the plastic hinging of the column. Maximum elastic moments within the positive and negative moment sections of the girders for the earthquake loads computed in Sections I8.3 and I8.4 are given in Table I-10. From Table I-10, one can see that load combinations consisting of 100 percent of the moment from the longitudinal earthquake load (MLEQ) plus 30 percent of the moment from the transverse earthquake load (MTEQ) will produce the largest elastic design moments. In addition, elastic design moments will be greatest for the interior girder. Interior girder moments due to 30 percent of the transverse earthquake load are small and can be ignored; therefore, elastic design moments for the positive and negative moment sections of the girder are as follows: + MELASTIC = 1,851 kN-m (1,365 k-ft) − MELASTIC = 3,242 kN-m (2,391 k-ft) Since girder moments from the transverse earthquake load are negligible, compare moments at the top of the col- umn from the longitudinal earthquake load, including a response modification factor of 1.0 rather than the value of 2.0 used to determine MMOD., and the moment associated with plastic hinging to determine the controlling condition for design. (ML)TOP = 12,932 kN-m (9,539 k-ft) (see Section I9.3.1) (MOVRSTR.)TOP = 17,184 kN-m (12,675 k-ft) (see Section I9.5) (ML)TOP < (MOVRSTR.)TOP Since the elastic moment at the top of the column from the longitudinal earthquake load is less than the moment resulting from plastic hinging, use the elastic moments computed above for the seismic design of the girders. Table I-11 compares the factored seismic girder moments to the factored live load moments used in the preliminary design of the girders. As seen in Table I-11, seismic moments are less than corresponding live load moments; there- fore, girder dimensions from the design for the Strength I limit state are adequate to resist the seismic moments devel- oped within the girders. I-36 Max. MLEQ kN-m (k-ft) Max. MTEQ kN-m (k-ft) Pos. Mom. Section ± 1851 (1365) ± 58 (43) Interior Girder Neg. Mom. Section ± 3242 (2391) ± 85 (63) Pos. Mom. Section ± 1631 (1203) ± 167 (123) Exterior Girder Neg. Mom. Section ± 1741 (1284) ± 263 (194) TABLE I-10 Maximum elastic seismic moments within the girders γMEQ kN-m (k-ft) γMLL kN-m (k-ft) Pos. Mom. Section +1851 (1365) +3866 (2852) Neg. Mom. Section -3242 (2391) -4914 (3625) TABLE I-11 Maximum factored girder moments

I10.2 Live Loading Check preliminary girder dimensions for the live load moments obtained from the computer model discussed in Sec- tion I6.1. Table I-12 compares the maximum unfactored live load girder moments obtained from the 3-dimensional computer model to those obtained from a line girder analysis program. As seen in Table I-12, the maximum moments from the computer model are less than the corresponding moments obtained from the line girder analysis program, therefore, the girder design for the Strength I limit state may be based on line girder analysis. This will allow the use of existing conventional computer programs to design the girders. I11 CAP BEAM DESIGN Initial dimensions of the pier cap were determined in Section I5.3 and are shown in Figure I-11. I11.1 Flexure I11.1.1 Factored Design Moment Design for the moment at the face of the column. Per Article 5.13.3.4, in the case of columns that are not rectan- gular, the critical section shall be taken at the side of the concentric rectangle of equivalent area. Column area = 2,630,220 mm2 (4,077 in2) From the loading diagrams shown in Figure I-14, calculate the dead load moments at the face of the equivalent column. MDC = 866(3.764) + 941(0.714) + 19.54(3,764)2/[2(1,000)2] = 4,070 kN-m (3,002 k-ft) MDW = 126(3.764) + 178(0.714) = 601 kN-m (443 k-ft) From the 3-dimensional computer model, the unfactored live load moment and corresponding torsion in the cap beam at the face of the column are given as follows: MLL = 3,201 kN-m (2,361 k-ft) TLL = 21 kN-m (15 k-ft) Factored moment and torsion Mu = 1.25(4,070) + 1.5(601) + 1.75(3,201) = 11,591 kN-m (8,550 k-ft) Tu = 1.75(21) = 37 kN-m (27 k-ft) The corresponding factored torsion is small and therefore can be neglected. Equivalent square column = 2,630,220 mm 1,622 mm (63.85 in. 63.85 in.)= × ×1 622, I-37 3-D Computer Model Line Girder Analysis Pos. Mom. Section +1844 (1360) +2209 (1629) Interior Girder Neg. Mom. Section -2634 (1943) -2808 (2071) Pos. Mom. Section +2226 (1642) +2577 (1901) Exterior Girder Neg. Mom. Section -3025 (2231) -3275 (2416) TABLE I-12 Maximum unfactored live load girder moments, kN-m (k-ft)

I11.1.2 Nominal Flexural Resistance Determine the nominal flexural resistance of the compression flange in accordance with Article 6.11.2.1.3a. For compression flanges without longitudinal stiffeners, w = b = 2,280 − 2(30) = 2,220 mm (87.40 in.) k = 4 Fn = 181,000 RbRhk(tf / w)2 Eq. 6.11.2.1.3a-3 Determine Rb from Article 6.10.4.3.2a. For box symmetrical about the horizontal axis, Dc = D/2 = 1,412/2 = 706 mm (27.80 in.) fc = Mu/S S = I / c = 8.520 × 1010/(1,472/2) = 1.157 × 108 mm3 (7,060 in3) Eq. 6.10.4.3.2a-1 Per Article 6.10.4.3.1a, Rh = 1.0 for homogeneous sections, Fn = 181,000(1.0)1.0(4)(30/2,220)2 = 132.2 Mpa (19.2 ksi) ϕFn = 1.0(132.2) = 132.2 MPa (19.2 ksi) > 100.2 MPa (14.5 ksi) applied stress ∴ OK Tension flange is OK by inspection. 2D t E f R c w b c b< ∴ =λ .1 0 λ b c E f = =5 76 200 000 100 2 257. , . f 11,591(1, 000)1.157 10 MPa (14.5 ksi)c 2 8= × = 100 2. 2D t c w > = 1 412 30 47 , Since w t kE Ff yc > 1 23. 1 23 1 23 4 200 000345 59 2. . ( , ) . kE Fyc = = 0 57 0 57 4 200 000345 27 4. . ( , ) . kE Fyc = = w tf = = 2 220 30 74 , I-38

I11.1.3 Web Slenderness Check web slenderness in accordance with Article 6.11.2.1.3b. Per Article 6.10.2.2, webs shall be proportioned such that Eq. 6.10.2.2-1 I11.2 Shear Design for the maximum shear force in the flange and web plates of the cap beam from either the Extreme Event I or Strength I load combinations. I11.2.1 Shear Forces for the Extreme Event I Load Combination Shear forces in the flange and web plates of the cap beam result from both flexure and torsion. Web plates experi- ence both flexural and torsional shears while flange plates see only torsional shear as shown in Figure I-22. Vu = Vf + VT For earthquake loads, design for the lesser of the elastic forces computed in accordance with Article 3.10.8 or the forces associated with the plastic hinging of the column. Maximum elastic forces in the pier cap for the earthquake loads computed in Sections I8.3 and I8.4 are given in Table I-13. Shear in the pier cap from the transverse earthquake load is small, therefore, design for the load combination con- sisting of 100 percent of the shear due to the forces from the longitudinal earthquake load plus 30 percent of the shear due to the forces from the transverse earthquake load. Thirty percent of the shear from the transverse earthquake load is small and can be ignored; therefore, the maximum elastic shear in the pier cap can be taken as that due to the tor- sion from the longitudinal earthquake load. 2D t E f OK c 2 c < ∴6 77. 6 77 6 77 200 000100 2 302 5. . , . . E fc = = 2D t c w = 47 2D t E f c w c ≤ ≤6 77 200. I-39 Torsional Shear Flexural Shear T VT P Vf Figure I-22. Cap beam shear forces.

Since the torsion in the cap beam at the face of the column is equal to half of the longitudinal moment in the column at the centerline of the pier cap, compare moments at the top of the column from the longitudinal earthquake load and plastic hinging to determine the controlling condition for design. (ML)TOP = 12,932 kN-m (9,539 k-ft) (see Section I9.3.1) (MOVRSTR.)TOP = 17,184 kN-m (12,675 k-ft) (see Section I9.5) (ML)TOP < (MOVRSTR.)TOP Since the elastic moment at the top of the column from the longitudinal earthquake load is less than the moment resulting from plastic hinging, design for the elastic shear due to the torsion from the longitudinal earthquake load. Calculate shear forces due to torsion. VT = qD Where, q = shear flow = T/(2Ao) (kN/m) Ao = enclosed area within the box section (mm2) D = width or depth of plate between webs or flanges (m) = 2.22 m (7.28 ft) for calculation of force in flange plates = 1.412 m (4.63 ft) for calculation of force in web plates Tu = TEQ = 8,632 kN-m (6,367 k-ft) Ao = (2,280 − 30)(1,472 − 30) = 3,244,500 mm2 (5,029 in2) (VT)FLANGE PLS. = qD = 1,330(2.220) = 2,953 kN (664 kips) (VT)WEB PLS. = qD = 1,330(1.412) = 1,878 kN (422 kips) Calculate shear forces per web plate due to dead load flexure, Vf. See Figure I-14 for dead loads. (Vf)DC = 1.25[866 + 941 + 19.54(4,575)/1,000]/2 = 1,185 kN (266 kips) (Vf)DW = 1.5(178 + 126)/2 = 228 kN (51 kips) Vf = (Vf)DL = 1,185 + 228 = 1,413 kN (318 kips) See Figure I-23 for the shear forces in the flange and web plates for the Extreme Event I load combination (i.e., due to seismic load plus dead load). I11.2.2 Shear Forces for the Strength I Load Combination As permitted in Article 6.11.2.2.1, maximum shear will be taken as the sum of the absolute values of the shears due to maximum flexure and maximum torsion. Calculate shear forces due to torsion. From the 3-dimensional computer model, the maximum unfactored live load torsion in the pier cap is 2,484 kN-m (1,832 k-ft). q kN/m (91 kip/ft)= =8 632 1 0002 3 244 500 1 330 2 , ( , ) ( , , ) , I-40 Longitudinal Earthquake Load Transverse Earthquake Load Torsion kN-m (k-ft) Flexural Shear kN (kips) Torsion kN-m (k-ft) Flexural Shear kN (kips) 8632 (6,367) 0 (0) 0 (0) 26 (6) TABLE I-13 Maximum elastic seismic pier cap forces

Tu = 1.75(2,484) = 4,347 kN-m (3,206 k-ft) (VT)FLANGE PLS. = 670(2.220) = 1,487 kN (334 kips) (VT)WEB PLS. = 670(1.412) = 946 kN (213 kips) Calculate factored shear forces in the web plates due to flexure. From the 3-dimensional computer model, the maximum unfactored live load shear in the pier cap is 1,506 kN (339 kips). Therefore, the vertical shear per web plate from dead plus live load, not including torsion, is as follows: Vf = 1.75(1,506)/2 + 1,185 + 228 = 2,731 kN (614 kips) See Figure I-23 for the shear forces in the flange and web plates for the Strength I load combination including torsion. I11.2.3 Nominal Shear Resistance Determine the nominal shear resistance for the plates of the cap beam in accordance with Articles 6.11.2.2.1 and 6.10.7.2. Interior diaphragm plates are present within the cap beam and have a stiffening effect on the plates of the pier cap. Since these diaphragm plates are spaced relatively wide with respect to the depth of the pier cap, any stiffening effect on the plates of the cap beam will be ignored. Therefore, treat the plates of the cap beam as unstiffened. Vn = CVp Eq. 6.10.7.2-1 Vp = 0.58FywDtw Eq. 6.10.7.2-2 For the flange plates, For unstiffened web panels, the buckling coefficient, k, may be taken equal to 5 (Equation 6.10.7.3.3a-8 with do >> D). 1 10 1 10 200 000 5345 59 2. . , ( ) . Ek Fyw = = D t 2,220/30 = 74 w = q kN/m (46 kip/ft)= =4 347 1 0002 3 244 500 670 2 , ( , ) ( , , ) I-41 1413 - 1878 = -465 kN (-105 kips) 2953 kN (664 kips) 1413 + 1878 = 3291 kN (740 kips) 2731 - 946 = 1785 kN (401 kips) 1487 kN (334 kips) 2731 + 946 = 3677 kN (827 kips) Extreme Event I Strength I 2953 kN (664 kips) 1487 kN (334 kips) Figure I-23. Shear forces on cap beam.

Eq. 6.10.7.3.3a-6 ϕVn = 1.0(10,661) = 10,661 kN (2,397 kips) > 2,953 kN (664 kips) ∴ OK For the web plates, Eq. 6.10.7.3.3a-5 therefore, OK I11.3 Check of Box-Beam Flanges for Combined Moment and Torsional Shear Evaluate the bottom flange plate of the cap beam for the interaction between moment and torsion for the Extreme Event I and Strength I limit states by investigating the ratio of the factored force to the factored resistance for both moment and torsion. I11.3.1 Extreme Event I Limit State Maximum factored torsion. Tu = 8,632 kN-m (6,367 k-ft) (see Section I11.2.1) Shear in bottom flange plate due to the maximum factored torsion. Vu = 2,953 kN (664 kips) (see Section I11.2.1) Corresponding factored moment at the face of the column. Mu = 1.25(4,070) + 1.5(601) = 5,989 kN-m (4,417 k-ft) (see Section I11.1.1) Compute the nominal moment resistance of the bottom flange plate. ϕV kN (1,906 kips) > 3,677 kN (827 kips),n = =1 0 1 0 0 58 345 1 412 301 000 8 476 . ( . )( . )( )( , )( ) , , D t Ek F C 1.0 w yw < ∴ =1 10. D t 1, 412/30 = 47.1 w = V CV 0.8(0.58)(345)(2,220)(30)1, 000 10,661 kN (2,397 kips)n p= = = C 1.1074 200, 000(5) 345= = 0 8. C 1.10D t Ek F w yw = Since Ek F D t Ek F , yw w yw 1 10 1 38. .< < 1 38 1 38 200 000 5345 74 3. . , ( ) . Ek Fyw = = I-42

ϕMn = ϕFnS Where, ϕFn = 132.2 MPa (19.2 ksi) (see Section I11.1.2) S = 1.157 × 108 mm3 (7,060 in3) (see Section I11.1.2) Since the ratio of the factored force to the factored resistance for both moment and shear is low, moment-torsion interaction will not be critical for the Extreme Event I limit state. I11.3.2 Strength I Limit State Maximum factored torsion. Tu = 4,347 kN-m (3,206 k-ft) (see Section I11.2.2) Shear in bottom flange plate due to the maximum factored torsion. Vu = 1,487 kN (334 kips) (see Section I11.2.2) From the 3-dimensional computer model, the unfactored live load moment corresponding to the live loading pro- ducing the maximum factored torsion is 1,098 kN-m (810 k-ft). Therefore, the corresponding factored moment is as follows: Mu = 1.25(4,070) + 1.5(601) + 1.75(1,098) = 7,911 kN-m (5,835 k-ft) Since the ratio of the factored force to the factored resistance for both moment and shear is low, moment-torsion interaction will not be critical for the Strength I limit state. I11.4 Fatigue Requirements for Webs Per Article 6.11.1, check the fatigue requirements for webs according to the provisions specified in Article 6.10.6. Check requirements for flexure in accordance with Article 6.10.6.3. 0 95 0 95 36 200 000345 137. . ( , )kE Fyw = = k 9 D D 9(2) 36 c 2 2 =   = = D tw = =1 412 30 47, / V V u nϕ = = 1 487 10 661 0 14 , , . M M u nϕ = = 7 911 15 296 0 52 , , . V V u nϕ = = 2 953 10 661 0 28 , , . M M 5,989 15,296 0.39 u nϕ = = ϕM kN-m (11,282 k-ft)n = × =1 0 132 2 1 157 101 000 15 296 8 2 . ( . )( . ) ( , ) , I-43

fcf ≤ Fyw Eq. 6.10.6.3-1 The factored flexural stress of the cap beam under the Strength I limit state does not exceed yield, therefore, the com- pressive stress in the flange under unfactored dead load plus fatigue load will be less than yield. This means that Equa- tion 6.10.6.3-1 will be satisfied and there is no need for further calculations. Since the web is unstiffened, the shear requirement of Article 6.10.6.4 does not apply. I11.5 Constructability Per Article 6.11.1, check the constructability of the cap beam according to the provisions specified in Article 6.11.5. Check the webs of the cap beam in accordance with Article 6.11.5.2. Eq. 6.10.3.2.2-1 fcw < Fyw The factored flexural stress of the cap beam under the Strength I limit state does not exceed yield, therefore, the com- pressive flexural stress in the web due to factored dead loads acting on the noncomposite section during construction will be less than yield. This means that Equation 6.10.3.2.2-1 will be satisfied and there is no need for further calculations. I11.6 Service Limit State Control of Permanent Deflections Per Article 6.11.1, check the Service limit state control of permanent deflections as specified in Article 6.11.7. Per Article C6.11.7, this check does not apply since the pier cap is a single box section. I11.7 Fatigue Check fatigue resistance of details in accordance with Article 6.6.1.2. γ(∆f) < (∆F)n Eq. 6.6.1.2.2-1 Detail categories for the various details of the pier cap are given in Figure I-24. For simplification of calculations, provide infinite fatigue life for details. Eq. 6.6.1.2.5-1 For infinite life, Detail Category C controls. From Table 6.6.1.2.5-3, for Detail Category C, (∆F)TH = 69 MPa (10 ksi). γ(∆f) = 0.75(∆M)FATIGUE/S ( (∆F) ) = 34.5 MPa (5.0 ksi)n = 12 69 ( (∆ ∆F) F)n TH= 12 0 9 0 9 200 000 1 25 36 47 3 6672 2 . . ( , )( . )( ) ( ) , , E k D t MPa (532 ksi) > F therefore, w yw α   = = f E k D t Fcw w yw≤   ≤0 9 2 . α D t 0.95 kE F therefore, w yw < , I-44

From the computer model, the range in moment experienced by the pier cap due to the passage of the fatigue load, (∆M)FATIGUE, is 1,095 kN-m (808 k-ft). This value is computed at the centerline of the column and does not include impact. γ(∆f) = 0.75(1.15)1,095(1,000)2/1.157 × 108 = 8.2 MPa (1.2 ksi) γ(∆f) < (DF)n ∴ OK I12 GIRDER-TO-CAP BEAM CONNECTION Figure I-5 shows the components of the girder-to-cap beam connection. They include a bolted double-angle con- nection for the transfer of shear from the web of the girder to the web of the box-beam pier cap and flange splice plates for the transfer of girder moments across the width of the pier cap. The flange splice plates also transfer the difference between the moment on either side of the pier cap to the pier cap in the form of torsional moment. I12.1 Bolted Double-Angle Connection Design the connection for the shear transferred from the web of the girder to the web of the cap beam. Per Article 6.13.2.1.1, design the connection as slip-critical since the connection is subject to impact and the reversal of forces. I12.1.1 Shear Forces Due to Unfactored Loadings Unfactored shear forces are greatest for the interior beam and are as follows: VDC = VDC1 + VDC2 = 376 + 94 = 470 kN (106 kips) (from line girder analysis) VDW = 89 kN (20 kips) (from line girder analysis) VLL = 517 kN (116 kips) (from computer model) For seismic loading, design the connection for the lesser of the elastic shear including the response modification fac- tor or the shear associated with the plastic hinging of the column. From Table 3.10.7.1-2, the response modification fac- tor for the connection is 1.0, therefore, the modified design shear is simply equal to the elastic shear from the earthquake I-45 Bolt holes for connection to beam (Cat. B) Shear Stud (Cat. C) Full penetration weld connecting plates of cap beam (Cat. B) Toe of fillet weld connecting diaphragm to cap beam (Cat. C) Figure I-24. Detail categories for fatigue.

loading. Shear in the girder webs at the edge of the pier cap resulting from the earthquake loads computed in Sections I8.3 and I8.4 are given in Table I-14. From Table I-14, shear is greatest in the interior girder. In addition, shear due to the transverse earthquake load is small and can be ignored. Since the shear due to the transverse earthquake load is negligible, the magnitude of the shear force in the web of the girder is directly related to the longitudinal moment at the top of the column resulting from the longitudinal earthquake load. Therefore, compare longitudinal moments at the top of the column from the longitudi- nal earthquake load and plastic hinging to determine the controlling condition for seismic design. (ML)TOP = 12,932 kN-m (9,539 k-ft) (see Section I9.3.1) (MOVRSTR.)TOP = 17,184 kN-m (12,675 k-ft) (see Section I9.5) (ML)TOP < (MOVRSTR.)TOP Since the longitudinal moment at the top of the column from the longitudinal earthquake load is less than the moment resulting from plastic hinging, use the elastic shear from the longitudinal earthquake load for the seismic design of the connection. VEQ = VLEQ = 165 kN (37 kips) I12.1.2 Slip Resistance Per Article 6.13.2.1.1, slip-critical connections shall be proportioned to prevent slip under Load Combination Service II. VSERV. II = VDC + VDW + 1.3VLL VSERV. II = 470 + 89 + 1.3(517) = 1,231 kN (277 kips) In accordance with Article 6.13.2.8, the nominal slip resistance of a bolt in a slip-critical connection shall be taken as: Rn = KhKsNsPt Eq. 6.13.2.8-1 Assuming 24 mm (1 in.) diameter ASTM A 325M (A 325) bolts in standard holes, double shear, and Class B surface conditions, I12.1.3 Shear Resistance I12.1.3.1 Design Force. Per Article 6.5.5, since seismic forces within the girders are based on the elastic moment at the top of the column and not on the formation of a plastic hinge, the connection may be assumed to behave as a bear- ing-type connection at the extreme event limit state. No. of Bolts = 6 bolts= 1 231205 , R = 205 kN/bolt (46 kips/bolt)n = ×( )1 0 0 5 2 205 10 1 000 3 . ( . )( ) , I-46 VLEQ kN (kips) VTEQ kN (kips) Interior Girder 165 (37.1) 2 (0.4) Exterior Girder 2 (0.4) 11 (2.5) TABLE I-14 Elastic seismic girder shears

VEQ < VLL, therefore, the extreme event limit state will not control. As specified in Article 6.13.2.1.1, slip-critical connections shall also be designed for shear and bearing at the strength limit state. In accordance with Article 6.13.1, design the connection at the strength limit state for not less than the larger of or 0.75ϕVn Vu = 1.25(470) + 1.5(89) + 1.75(517) = 1,626 kN (366 kips) In accordance with Article 6.10.7.3.3b, the nominal shear resistance for the interior web panel of a noncompact sec- tion is determined as follows: From the preliminary design of the girders, fu = ϕfFy, therefore, Eq. 6.10.7.3.3b-2 for which, Eq. 6.10.7.3.3b-3 fu = ϕfFy, therefore, R = 0.6 Eq. 6.10.7.3.3a-8 Since the design of the transverse stiffeners is not presented, take do = 1,350 mm (53.15 in.) for this example. D t 1,372 12 114.3w = = 1 38 1 38 200 000 10 16345 105 9. . , ( . ) . Ek Fy = = 1 10 1 10 200 000 10 16345 84 4. . , ( . ) . Ek Fy = = k 5 5 1,350 1,372 2= +   = 10 16. k 5 5 d D o 2= +   R 0.6 0.4 F f F 0.75 F r u r f y = + − −        ϕ V RV C 0.87(1 C) 1 d D CVn p o 2 p= + − +         ≥ V V 2 u n+ ϕ I-47

Therefore, Eq. 6.10.7.3.3a-7 ϕVn = 1,740 kN (391 kips) 0.75ϕVn = 0.75(1,740) = 1,305 kN (293 kips) I12.1.3.2 Nominal Bolt Resistance. In accordance with Article 6.13.2.7, the nominal shear resistance of a high- strength bolt at the strength limit state in joints whose length between extreme fasteners measured parallel to the line of action of the force is less than 1,270 mm (50 in.) shall be taken as follows. Conservatively assuming bolt threads are included in the shear plane, Rn = 0.38AbFubNs Eq. 6.13.2.7-2 ϕRn = 0.8(285) = 228 kN/bolt (51 kips/bolt) I12.1.4 Bearing Resistance Check bearing on the connected material in accordance with Article 6.13.2.9. Bearing on web of girder controls. Minimum spacing = 3d = 3(24) = 72 mm (2.83 in.) Lc = 72 − 26 = 46 mm (1.81 in.) 2d = 2(24) = 48 mm (1.89 in.) No. of bolts = 7.4 bolts use 8 bolts1 683228 , = → R = 285 kN/bolt (64 kips/bolt)n 0 38 24 830 24 1 000 2 . ( ) ( )( ) ( , ) π = V = larger of V V and 0.75 V kN (378 kips)DSGN u n n+ =ϕ ϕ2 1 683, V V kN (378 kips)u n+ = + =ϕ2 1 626 1 740 2 1 683 , , , ϕVn = + − +         1 0 0 6 0 58 345 1 372 12 0 685 0 87 1 0 685 1 1 3501 372 1 0002. ( . )( . )( )( , )( ) . . ( . ) , , / , C =    = 1 52 114 3 200 000 10 16 345 0 6852 . ( . ) , ( . ) . C 1.52 D t Ek F w 2 y =       D t Ek Fw y > 1 38. I-48

Lc < 2d, therefore, Rn = 1.2LctFu Eq. 6.13.2.9-2 ϕRn = 0.8(321) = 257 kN/bolt (58 kips/bolt) ϕRn > Ru ∴ OK I12.1.5 Size Angles Assume Fy = 250 MPa (36 ksi) for angles. With the aid of design tables or through a series of short hand calculations, one can determine that an angle thick- ness of 12.7 mm (1/2 in.) will satisfy design requirements. In order to meet minimum edge distance requirements and still satisfy entering and tightening clearances, provide 127-mm (5-in.) angle legs. Space bolts at the maximum spacing allowed for sealing bolts as specified in Article 6.13.2.6.2, but do not exceed a connection length of 1,270 mm (50 in.), otherwise, the bolt resistance should be reduced in accordance with Arti- cle 6.13.2.7. s ≤ (100 + 4.0t) ≤ 175 Eq. 6.13.2.6.2-1 s ≤ [100 + 4(12.7)] = 150.8 mm (6.03 in.) s ≤ 150.8 mm (6.03 in.) → say s = 150 mm (6 in.) connection length = 150(8 − 1) = 1,050 mm (42 in.) < 1,270 mm (50 in.) ∴ OK I12.2 Flange Splice Plates Flange splice plates provide moment continuity for the girders. These plates also transfer torsion to the pier cap pro- duced by the difference in moments in the girders at either side of the pier cap. I12.2.1 Girder Moments Due to Unfactored Loadings Unfactored moments in the girders at the centerline of the pier cap are given in Table I-15. R = VNo. of bolts = 10 kN/bolt (47 kips/bolt)u DSGN = 1 683 8 2 , R = 3 kN/bolt (72 kips/bolt)n 1 2 46 12 4851 000 21 . ( )( )( ) , = I-49 MDC1a kN-m (k-ft) MDC2a kN-m (k-ft) MDWa kN-m (k-ft) MLLb kN-m (k-ft) MEQb kN-m (k-ft) Interior Girder 2827 (2085) 663 (489) 628 (463) 2634 (1943) 3242 (2391) Exterior Girder 2538 (1872) 663 (489) 445 (328) 3025 (2231) 1733 (1278) a From line girder analysis b From computer model TABLE I-15 Unfactored girder moments at centerline of pier

I12.2.2 Size Flange Plates I12.2.2.1 Design Force. Design flange splice plates for the Extreme Event I or Strength I limit state. MEQ < 1.75MLL, therefore, extreme event limit state will not control. Since the bolted double-angle connection to the web of the girder is assumed to possess little-to-no rotational restraint, moments at the point of splice are resisted entirely by the flange splice plates. Therefore, in accordance with Article 6.13.1, design splice plates at the strength limit state for a moment not less than the larger of or 0.75ϕMn Since moments are greatest at the centerline of the cap beam, it is conservative to design for the factored moment at the centerline of the pier cap instead of for the moments in the girders at the face of the pier cap. (MCL)INT. BM. = 1.25(2,827 + 663) + 1.5(628) + 1.75(2,634) = 9,914 kN-m (7,313 k-ft) (MCL)EXT. BM. = 1.25(2,538 + 663) + 1.5(445) + 1.75(3,025) = 9,963 kN-m (7,349 k-ft) (MCL)EXT. BM. > (MCL)INT. BM. ∴ Mu = 9,963 kN-m (7,349 k-ft) Slab reinforcing has been neglected in computing section properties for the design of the girders for negative moment. Therefore, ϕMn = ϕFnSnc Where Snc = section modulus for noncomposite steel section (mm3) = Inc/c Inc = 2.180 × 1010 mm4 (52,375 in4) (conservatively calculated for the gross section) Compute design moment. 0.75ϕMn = 0.75(10,219) = 7,664 kN-m (5,653 k-ft) To find the force in the flange splice plates, divide the design moment by the distance between the centers of the splice plates. Since the thickness of the plates is not yet known, divide the design moment by the cap beam depth to determine the force in the flange splice plates. PDSGN = 10,091/1.472 = 6,855 kN (1,541 kips) I12.2.2.2 Plate Thickness. Design for tension in accordance with Article 6.13.5.2. For gross section yield, M larger of M M2 and 0.75 M kN-m (7, 443 k-ft)DSGN u n n= + = ϕ ϕ 10 091, M M kN-m (7, 443 k-ft)u n+ = + =ϕ2 9 963 10 219 2 10 091 , , , ϕ ϕM F S kN-m (7,538 k-ft)n n nc= = ×( )    = ( . ) . ( / ) / , , 1 0 345 2 180 10 1472 2 1 000 10 219 10 2 M M 2 u n+ ϕ I-50

ϕPn = ϕyFyAg = ϕyFybt Eq. 6.8.2.1-1 ϕPn > PDSGN Substituting Equation 6.8.2.1-1 for ϕPn and solving for t gives t > 55 mm (2.17 in.) For net section fracture, ϕPn = ϕuFuAnU Eq. 6.8.2.1-2 Per Article 6.13.5.2, An ≤ 0.85Ag Assuming four 24-mm (1-in.) diameter bolts per section, An = [b − 4(28)]t < 0.85bt Dividing both sides of the inequality by t gives [380 − 4(28)] = 268 mm (10.55 in.) < 0.85(380) = 323 mm (12.72 in.), therefore, An controls ϕPn = ϕuFu[b − 4(28)]tU ϕPn > PDSGN Substituting for ϕPn and solving for t gives t > 65.9 mm (2.59 in.) → use t = 70 mm (2.75 in.) I12.2.3 Design Connection to Girder Flanges As specified in Article 6.13.6.1.4a, bolted splices for flexural members shall be designed using slip-critical connections. I12.2.3.1 Slip Resistance. Per Article 6.13.2.1.1, slip-critical connections shall be proportioned to prevent slip under Load Combination Service II. (MSERV. II)INT. BM. = 2,827 + 663 + 628 + 1.3(2,634) = 7,542 kN-m (5,563 k-ft) (MSERV. II)EXT. BM. = 2,538 + 663 + 445 + 1.3(3,025) = 7,579 kN-m (5,590 k-ft) (MSERV. II)EXT. BM. > (MSERV. II)INT. BM. ∴ MSERV. II = 7,579 kN-m (5,590 k-ft) PSERV. II = 7,579/1.472 = 5,149 kN (1,158 kips) t P F b U 6,855(1, 000) 0.8(485) 380 - 4(28) mm (2.59 in.) DSGN u u ≥ −[ ] = [ ] =ϕ 4 28 1 0 65 9( ) . . t P F b 6,855(1, 000) 0.95(345)(380) mm (2.17 in.) DSGN y y ≥ = =ϕ 55 I-51

In accordance with Article 6.13.2.8, the nominal slip resistance of a bolt in a slip-critical connection shall be taken as: Rn = KhKsNsPt Eq. 6.13.2.8-1 Assuming 24-mm (1-in.) diameter ASTM A 325M (A 325) bolts in standard holes, single shear, and Class B surface conditions, I12.2.3.2 Shear Resistance. In accordance with Article 6.13.2.7, the nominal shear resistance of a high-strength bolt at the strength limit state in joints whose length between extreme fasteners measured parallel to the line of action of the force is less than 1,270 mm (50 in.) shall be taken as follows. For a 50-mm (2-in.) thick girder flange, bolt threads will be excluded from the shear plane. Therefore, Rn = 0.48AbFubNs Eq. 6.13.2.7-1 ϕRn = 0.8(180.2) = 144.2 kN/bolt (32.4 kips/bolt) I12.2.3.3 Bearing Resistance. Check bearing on the connected material in accordance with Article 6.13.2.9. Bearing on flange of girder controls. From Table 6.13.2.6.6-1, for a 24-mm (1-in.) diameter bolt and a rolled- or gas-cut edge, minimum end distance equals 30 mm (1.25 in.). Therefore, Lc = 30 − 26/2 = 17 mm (0.67 in.) 2d = 2(24) = 48 mm (1.89 in.) Lc < 2d, therefore, Rn = 1.2LctFu Eq. 6.13.2.9-2 ϕRn = 0.8(494.7) = 395.8 kN/bolt (89.0 kips/bolt) ϕRn > Ru ∴ OK I12.2.4 Design Connection to Cap Beam I12.2.4.1 Design Force. Design the connection for the transfer of torsion to the cap beam resulting from the difference in girder moments. Torsion is transferred to the pier cap through the flange splice plates and their connection to the pier cap as shown in Figure I-6. From Figure I-6, the design force for the connection can be taken as follows. R PNo. of bolts kN/bolt (29.7 kips/bolt)u DSGN = = = 6 855 52 132 , R kN/bolt (111.2 kips/bolt)n = =1 2 17 50 4851 000 494 7 . ( )( )( ) , . No. of bolts = 6,855144.2 bolts say 48 bolts < 52 bolts does not control= → ∴47 5. R = 0.48 (24)4(1, 000) kN/bolt (40.5 kips/bolt)n 2π ( )( ) . 830 1 180 2= No. of Bolts = 5,149102.5 50.2 bolts use 13 rows of 4 bolts = 52 bolts= → R kN/bolt (23.0 kips/bolt)n = ×( ) = 1 0 0 5 1 205 10 1 000 102 5 3 . ( . )( ) , . I-52

V = T/d Where d = depth of pier cap (m) The torsion transferred to the pier cap through the flange splice plates can be taken as the change in torque at the girder location from the torsion diagram for the pier cap. Unfactored torsion diagrams for the pier cap for earthquake loading and the live load case that produces the maximum difference in girder moment (and, consequentially, maxi- mum torsion transfer to the pier cap) from one side to another of the pier cap are given in Figure I-25. The pier cap sees no torsion from the transverse earthquake load or dead load, therefore, torsion diagrams are not shown in Figure I-25 for these loads. For the Extreme Event I limit state, VEXTR. EVENT I = TEQ/d = 5,493/1.472 = 3,732 kN (839 kips) In accordance with Article 6.13.1, at the strength limit state, the connection shall be designed for not less than the larger of V V 2 u n+ ϕ I-53 a) Longitudinal Earthquake Load b) Live Load pier cap 2,031 kN-m (1,498 k-ft) 2,484 kN-m (1,832 k-ft) 453 kN-m (334 k-ft) 3,138 kN-m (2,315 k-ft) 5,493 kN-m (4,052 k-ft) 8,631 kN-m (6,366 k-ft) pier cap Figure I-25. Unfactored torsion diagrams for pier cap.

or 0.75ϕVn Vu = 1.75TLL/d = 1.75(2,031)/1.472 = 2,415 kN (543 kips) Base the factored resistance on the tensile strength of the flange splice plates in accordance with Article 6.13.5.2. For gross section yield, ϕPn = ϕyFyAg Eq. 6.8.2.1-1 For net section fracture, ϕPn = ϕuFuAnU Eq. 6.8.2.1-2 The girder moment corresponding to the factored resistance of the flange splice plates is as follows: M = ϕPn(d + tSPLICE PL.) = 7,279(1.472 + 0.070) = 11,224 kN-m (8,279 k-ft) For the loading that produced the live load torsion diagram shown in Figure I-25, the maximum unfactored nega- tive moment in the exterior girder is 2,224 kN-m (1,640 k-ft). This corresponds to an increase in moment at the exte- rior girder of 11,224/2,224 = 5.047 Since the torsion in the pier cap results from the moments in the girder, the torsion in the pier cap at this location is increased by the same factor. Therefore, when the factored resistance of the flange splice plates is reached, the corre- sponding torsion in the pier cap will be as follows: T = 5.047(2,031) = 10,250 kN-m (2,304 k-ft) ϕVn = T/d = 10,250/1.472 = 6,963 kN (1,565 kips) 0.75ϕVn = 0.75(6,963) = 5,222 kN (1,174 kips) VEXTR. EVENT I = 3,732 kN (839 kips) < VSTR. I = 5,222 kN (1,174 kips), therefore, VDSGN = 5,222 kN (1,174 kips) I12.2.4.2 Shear Resistance. For a 24-mm (1-in.) diameter ASTM A 325M (A 325) bolt and a connection length less than 1,270 mm (50 in.), ϕRn = 144.2 kN/bolt (32.4 kips/bolt) (see Section I12.2.3.2) V larger of V V and 0.75 V kN (1,174 kips)STR. I u n n= + =ϕ ϕ2 5 222, V V 2 kN (1, 054 kips) u n+ = + = ϕ 2 415 6 963 2 4 689 , , , ϕP 0.8(485) 380 - 4(28)1, 000 kN (1,636 kips) n = [ ] = ← ( )( . ) , 70 1 0 7 279 controls ϕP 0.95(345)(380)(70)1, 000 kN (1,960 kips)n = = 8 718, I-54

However, since the width of the pier cap is 2,280 mm (89.76 in.), the connection length will be greater than 1,270 mm (50 in.). Therefore, in accordance with Article 6.13.2.7, reduce the bolt resistance by a factor of 0.80. ϕRn = 0.8(144.2) = 115.4 kN/bolt (25.9 kips/bolt) I12.2.4.3 Slip Resistance. Per Article 6.13.2.1.1, slip-critical connections shall be proportioned to prevent slip under Load Combination Service II. VSERV. II = 1.3TLL/d = 1.3(2,031)/1.472 = 1,794 kN (403 kips) Assuming 24-mm (1-in.) diameter ASTM A 325M (A 325) bolts in standard holes and Class B surface conditions, Rn = 102.5 kN/bolt (23.0 kips/bolt) (see Section I12.2.3.1) I12.2.4.4 Bearing Resistance. Bearing on flange plate of cap beam controls. Minimum spacing = 3d = 3(24) = 72 mm (2.83 in.) Lc = 72 − 26 = 46 mm (1.81 in.) 2d = 2(24) = 48 mm (1.89 in.) Lc < 2d, therefore, Rn = 1.2LctFu Eq. 6.13.2.9-2 ϕRn = 0.8(803) = 642 kN/bolt (144 kips/bolt) ϕRn > Ru ∴ OK I12.3 Girder-to-Cap Beam Connection Details Refer to Figure I-26 for final girder-to-cap beam connection details. I13 COLUMN-TO-CAP BEAM CONNECTION The transfer of forces between the column and cap beam is achieved through the use of shear studs located as shown in Figure I-27. R VNo. of bolts kN/bolt (25.5 kips/bolt)u DSGN = = = 5 222 46 113 5 , . R 1.2(46)(30)(485)1, 000 kN/bolt (181 kips/bolt)n = = 803 No. of Bolts = 1, 794102.5 bolts say 18 bolts < 46 bolts does not control= → ∴17 5. No. of Bolts = 1, 794102.5 bolts say 18 bolts < 46 bolts does not control= → ∴17 5. I-55

I13.1 Shear Studs on Bottom Flange Plate Shear studs located on the bottom flange plate of the cap beam shall be designed to transfer horizontal shear between the column and cap beam. I13.1.1 Strength Design I13.1.1.1 Design Force. Design these studs for the maximum horizontal shear, H, developed at the top of the column. HEXTR. EVENT I = HEQ = 5,349 kN (1,203 kips) (see Section I9.6.1) HSTR. I = 1.75(HLL) = 1.75(506) = 886 kN (199 kips) (from computer model) HEXTR. EVENT I > HSTR. I ∴ HDSGN = 5,349 kN (1,203 kips) I-56 Section A-A 22 spaces @ 95 mm (3.75 in.)12 spaces @ 75 mm (3 in.) 12 spaces @ 75 mm (3 in.) 75 mm (3 in.) 24 mm (1 in.) dia. A325M (A325) bolt 30 mm (1.25 in.) L127x127x12.7 (L5"x5"x1/2") Fy = 250 MPa (36 ksi) 7 spa @ 150 mm (6 in.) = 1050 mm (42 in.) 50 mm (2 in.) AA 24 mm (1 in.) dia. A325M (A325) bolt 70 mm (2.75 in.) thick splice plate ASTM A709M (A709), Grade 345W (50W) Internal diaphragm 50 mm (2 in.) 75 mm (3 in.) 30 mm (1.25 in.) Provide 2 additional bolts to seal edge against the penetration of moisture Figure I-26. Girder-to-cap beam connection details.

Notice that due to the symmetric geometry of the structure, dead loads do not produce shear in the column. I13.1.1.2 Nominal Shear Resistance. In accordance with Article 6.10.7.4.4c, the nominal shear resistance for one shear stud shall be taken as Eq. 6.10.7.4.4c-1 Assuming 25-mm (1-in.) diameter studs, Asc = π(25)2/4 = 491 mm2 (0.76 in2) AscFu = 491(415)/1,000 = 203.8 kN/stud (45.8 kips/stud) Qn = 203.8 kN/stud (45.8 kips/stud) As specified in Article 1.3.2.1, ϕsc = 1.0 for extreme event limit states. Q kN/stud (46.5 kips/stud) > 203.8 kN/stud (45.8 kips/stud), therefore,n = =0 5 491 28 25 3991 000 207 . ( ) ( , ) , E f MPa (3,684 ksi)c c= ′ = =4 800 4800 28 25 399, , Q 0 A f E A Fn sc c c sc u= ′ ≤.5 I-57 A A Transfer horizontal shear between column and cap beam Section A-A ML MT Carry shear from beams to column produced by MT Carry shear from beams to column produced by ML. Also carry shear from axial load since this is the most direct load path to the column from the beams. B B D D C C diaphragm girder See Figures I-28, I-29, and I-30 for Sections B-B, C-C, and D-D, respectively. Figure I-27. Column-to-cap beam connection.

ϕscQn = 1.0(203.8) = 203.8 kN/stud (45.8 kips/stud) No. of studies = 5,349/203.8 = 26.2 studs → use 27 studs Per Article 6.10.7.4.1a, the ratio of the height to the diameter of a shear stud shall not be less than 4. h/d > 4 h > 4d = 4 (25) = 100 mm (4 in.) I13.1.2 Fatigue Design Fatigue of the shear studs is not a concern since the live load shear is considerably less than the design level shear force due to the earthquake load. I13.1.3 Shear Stud Layout For strength, provide a minimum of twenty-seven 25-mm (1-in.) diameter shear studs at least 100 mm (4 in.) in length on both the top and bottom side of the bottom flange plate of the cap beam. Refer to Figure I-28 for the shear stud layout on the bottom flange plate of the cap beam. I-58 Pocket in column for shear studs. Fill with grout after cap beam is placed. 150 mm (6 in.) dia. hole in bottom plate for placing grout. 25 mm (1 in.) dia. x 150 mm (6 in.) shear stud. Both sides of plate. 6 spa @ 150 mm (6 in.) 4 sp a @ 1830 mm (6 ft) dia. column Bottom plate of cap beam Section B-B See Figure I-27 for location 15 0 m m (6 in . ) Figure I-28. Stud layout for bottom flange plate of cap beam.

I13.2 Shear Studs on Web Plates of Cap Beam Shear studs located on the web plates of the cap beam shall be designed to transfer shear from the girder webs to the top of the column. These studs shall be designed for the shear originating from the longitudinal moment at the top of the column. In addition, since the most direct load path from the beams to the column is through the web plates of the cap beam (as opposed to the diaphragm plates), these studs shall also be designed to carry the shear resulting from the axial load in the column. These shear forces are shown in Figure I-8 and their magnitude per web plate is determined as follows: VAXIAL = P / 2 VLONG. MOM. = ML / w Where w = width of pier cap less web plates (m) = 2.220 m (7.28 ft) I13.2.1 Strength Design I13.2.1.1 Shear Forces for Extreme Event I Limit State. Shear due to axial dead load. VDL = PDL / 2 PDL = 1.25PDC + 1.5PDW PDL = 1.25(3,880) + 1.5(608) = 5,762 kN (1,295 kips) see Section I9.1 for PDC & PDW VDL = 5,762/2 = 2,881 kN (648 kips) Shear due to longitudinal moment at the top of the column from earthquake loading. VEQ = (ML)EQ / w In accordance with Article 3.10.9.4.1, (ML)EQ shall be taken as the lesser of MMOD. = MELASTIC /R or MOVRSTR. Per Article 3.10.8, MELASTIC = 1.0(12,932) + 0.3(0.0) = 12,932 kN-m (9,539 k-ft) See Table I-6 for values. From Table 3.10.7.1-2, for a column-to-cap beam connection, R = 1.0. MMOD. = 12,932/1.0 = 12,932 kN-m (9,539 k-ft) MOVRSTR. = 17,184 kN-m (12,675 k-ft) (see Section I9.5) (ML)EQ = lesser of MMOD. and MOVRSTR. = 12,932 kN-m (9,539 k-ft) I-59

VEQ = 12,932/2.220 = 5,825 kN (1,310 kips) VEXTR. EVENT I = VDL + VEQ = 2,881 + 5,825 = 8,706 kN (1,957 kips) I13.2.1.2 Shear Forces for Strength I Limit State. Shear force for Load Case 1 (maximum axial load) PLL = 1.75(2,465) = 4,314 kN (970 kips) (from Table I-7) Corresponding longitudinal moment at the top of the column (ML)LL = 1.75(25) = 44 kN-m (32 k-ft) (from Table I-7) (VSTR. I)LC1 = (5,762 + 4,314)/2 + (44/2.220) = 5,058 kN (1,137 kips) Shear force for Load Case 2 (maximum longitudinal moment) (ML)LL = 1.75(2,489) = 4,356 kN-m (3,213 k-ft) (from Table I-7) Corresponding axial load at the top of the column PLL = 1.75(1,218) = 2,132 kN (479 kips) (from Table I-7) (VSTR. I)LC2 = (5,762 + 2,132)/2 + (4,356/2.220) = 5,909 kN (1,328 kips) controls Determine controlling limit state for strength design. ϕsc = 0.85 for strength limit state VSTR. I / ϕsc = 5,909/0.85 = 6,952 kN (1,563 kips) < VEXTR. EVENT I = 8,706 kN (1,957 kips), therefore, VDSGN = VEXTR. EVENT I = 8,706 kN (1,957 kips) I13.2.1.3 Nominal Shear Resistance. Assuming 25-mm (1-in.) diameter shear studs, ϕscQn = 203.8 kN/stud (45.8 kips/stud) (see Section I13.1.1.2) No. of studs = 8,706/203.8 = 42.7 studs → use 43 studs I13.2.2 Fatigue Design I13.2.2.1 Live Load Shear Force Range. Shear force range for Load Case 1 (maximum axial load) (∆P)FATIGUE = 0.75(1.15)(314) = 271 kN (61 kips) (from computer model) Corresponding longitudinal moment at the top of the column (∆ML)FATIGUE = 0.75(1.15)(52) = 45 kN-m (33 k-ft) (from computer model) (Vsr)LC1 = (271)/2 + (45/2.220) = 156 kN (35 kips) Shear force range Load Case 2 (maximum longitudinal moment) (∆ML)FATIGUE = 0.75(1.15)(461) = 398 kN-m (294 k-ft) (from computer model) I-60

Corresponding axial load at the top of the column (∆P)FATIGUE = 0.75(1.15)(203) = 175 kN (39 k-ft) (from computer model) (Vsr)LC2 = (175)/2 + (398/2.220) = 267 kN (60 k-ft) ← controls I13.2.2.2 Fatigue Resistance. In accordance with Article 6.10.7.4.2, the fatigue resistance of an individual shear con- nector, Zr, shall be taken as Zr = αd2 ≥ 38d2/2 Eq. 6.10.7.4.2-1 For simplification of calculations, No. of studs = 267/11.9 = 22.4 studs → say 23 studs < 43 provided, therefore, does not control. I13.2.3 Shear Stud Layout For strength, provide a minimum of forty-three 25-mm (1-in.) diameter shear studs at least 100 mm (4 in.) in length on each web plate of the cap beam within the joint region of the pier cap. Refer to Figure I-29 for the shear stud lay- out on the web plates of the cap beam. I13.3 Shear Studs on Diaphragm Plates Shear studs located on the diaphragm plates adjacent to the joint region within the cap beam shall be designed to transfer shear from the girder webs to the top of the column. These studs shall be designed for the shear originating Z d kN/stud (2.7 kips/stud)r = = =382 38 25 2 1 000 11 9 2 2( ) ( , ) . I-61 8 spa @ 300 mm (11 13/16") 4 sp a @ CL Column 25 mm (1 in.) dia. x 150 mm (6 in.) shear stud Section C-C See Figure I-27 for location 30 0 m m (11 13 /1 6" ) Figure I-29. Stud layout for web plates of cap beam.

from the transverse moment at the top of the column. This shear force is shown in Figure I-9 and can be determined as follows: VTRANSV. MOM. = MT / S Where S = spacing between interior beams (m) = 3.050 m (10 ft) I13.3.1 Strength Design I13.3.1.1 Shear Force for Extreme Event I Limit State. Shear due to transverse moment at the top of the column from earthquake loading. VEQ = (MT)EQ /S (MT)EQ = lesser of MMOD. and MOVRSTR. MMOD. = MELASTIC /R MMOD. = [1.0(967) + 0.3(0.0)]/1.0 (moment values from Table I-6) MMOD. = 967 kN-m (713 k-ft) MOVRSTR. = 17,184 kN-m (12,675 k-ft) (see Section I9.5) (MT)EQ = lesser of MMOD. and MOVRSTR. = 967 kN-m (713 k-ft) VEQ = 967/3.05 = 317 kN (71 kips) VEXTR. EVENT I = VEQ = 317 kN (71 kips) I13.3.1.2 Shear Force for Strength I Limit State. (MT)LL = 1.75(3,417) = 5,980 kN-m (4,411 k-ft) (from Table I-7) VSTR. I = VLL = 5,980 / 3.05 = 1,961 kN (441 kips) ← controls I13.3.1.3 Nominal Shear Resistance. Assuming 25-mm (1-in.) diameter shear studs, Qn = 203.8 kN/stud (45.8 kips/stud) (see Section I13.1.1.2) ϕscQn = 0.85(203.8) = 173.2 kN/stud (38.9 kips/stud) No. of studs = 1,961/173.2 = 11.3 studs → say 12 studs I13.3.2 Fatigue Design Live Load Shear Force Range. (∆MT)FATIGUE = 0.75(1.15)(1,037) = 894 kN-m (659 k-ft) (from computer model) Vsr = 894 / 3.05 = 293 kN (66 kips) I-62

Fatigue Resistance Zr = 11.9 kN/stud (2.7 kips/stud) (see Section I13.2.2.2) No. of studs = 293/11.9 = 24.6 studs → use 25 studs I13.3.3 Shear Stud Layout For fatigue, provide a minimum of twenty-five 25-mm (1-in.) diameter shear studs at least 100 mm (4 in.) in length on each diaphragm on each side of the joint region. Refer to Figure I-30 for the shear stud layout on the diaphragm plates adjacent to the joint region within the pier cap. I-63 5 spa @ 300 mm (11 13/16") 5 sp a @ Section D-D See Figure I-27 for location Access hole used during assembly. Cover before placing concrete in joint region of cap beam. outline of cover plate 25 mm (1 in.) dia. x 150 mm (6 in.) shear stud 22 0 m m (8 11 /16 " ) Figure I-30. Shear stud layout for diaphragm plates adjacent to joint region.